SSAT Middle Level Quiz: Exponent Expressions

20 questions · exam conditions

0:00

Exponent ExpressionsQuestion 1 of 20

A star is $6.2\times10^4$ km away; base $10$ , exponent $4$ : value?

62,000

6,200

620,000

6,020

← Back to quizzes

SSAT Middle Level Quiz

SSAT Middle Level Quiz: Exponent Expressions

Practice Exponent Expressions in SSAT Middle Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Exponent Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for SSAT Middle Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A star is $6.2\times10^4$ km away; base $10$ , exponent $4$ : value?

62,000 (correct answer)
6,200
620,000
6,020

Explanation: This question tests the SSAT Middle Level skill of evaluating expressions with exponents, focusing on understanding the properties of exponents. Exponents represent repeated multiplication of a base number, and understanding this helps in evaluating expressions accurately. In this specific question, the expression 6.2×10^4 is used to illustrate scientific notation, where 10^4 means 10,000, resulting in 62,000. The correct choice, 62,000, shows the accurate application of the exponent rule which states that the base 10 is multiplied by itself four times. A common distractor might be miscounting zeros, such as 6,200, demonstrating a misunderstanding of place value. Teaching strategies include practicing exponent rules through repeated exercises and using real-world examples like astronomical distances to solidify understanding. Encourage students to visualize exponentiation as multi-step multiplication rather than simple addition.

Question 2

What is the value of $(2^3)^2 \cdot (2^2)^3$ ?

$2^{10}$
$2^{11}$
$2^{12}$ (correct answer)
$4^{11}$
$8^6$

Explanation: When you see expressions with exponents raised to other exponents, you're working with the power rule of exponents. This rule states that

(a^m)^n = a^{m \cdot n}

- you multiply the exponents together. Let's break down

(2^3)^2 \cdot (2^2)^3

step by step. First, apply the power rule to each part separately. For

(2^3)^2

, multiply the exponents:

2^{3 \cdot 2} = 2^6

. For

(2^2)^3

, do the same:

2^{2 \cdot 3} = 2^6

. Now you have

2^6 \cdot 2^6

. When multiplying powers with the same base, you add the exponents:

2^6 \cdot 2^6 = 2^{6+6} = 2^{12}

. Looking at the wrong answers: Choice (A)

2^{10}

results from incorrectly adding exponents in the first step (getting

2^5 \cdot 2^5

) instead of multiplying them. Choice (B)

2^{11}

might come from miscalculating somewhere in the middle steps - perhaps getting

2^5 \cdot 2^6

and then correctly adding those exponents. Choice (D)

4^{11}

represents a fundamental confusion about bases and exponents, possibly thinking

2^2 = 4

means you should change the entire expression to base 4. The answer is (C)

2^{12}

. Remember this pattern:

(a^m)^n = a^{mn}

and

a^m \cdot a^n = a^{m+n}

. Many SSAT exponent problems combine these two rules, so practice identifying when to multiply exponents versus when to add them.

Question 3

If $4^x = 64$ , what is the value of $2^x$ ?

4
8 (correct answer)
16
32
64

Explanation: When you encounter equations with exponents where the bases are related, look for ways to express both sides using the same base to make the problem easier to solve. Starting with

4^x = 64

, notice that both 4 and 64 can be written as powers of 2. Since

4 = 2^2

and

64 = 2^6

, you can rewrite the equation as

(2^2)^x = 2^6

. Using the power rule for exponents,

(2^2)^x = 2^{2x}

, so the equation becomes

2^{2x} = 2^6

. When the bases are equal, the exponents must be equal:

2x = 6

, which gives you

x = 3

. Therefore,

2^x = 2^3 = 8

. Looking at the wrong answers: Choice (A) 4 might tempt you if you confused

2^x

with the base 4 from the original equation, but these are completely different expressions. Choice (C) 16 equals

2^4

, which you'd get if you mistakenly thought

x = 4

- this could happen if you incorrectly solved

4^x = 64

and got the wrong value for

x

. Choice (D) 32 equals

2^5

, suggesting

x = 5

, which might result from computational errors when converting to the same base or solving for

x

. Remember this strategy: when dealing with exponential equations, try to express everything using the same base, especially when you see numbers that are clearly powers of 2 (like 4, 8, 16, 32, 64). This technique transforms complex-looking problems into straightforward algebra.

Question 4

Which expression has the same value as $\frac{6^4}{6^2}$ ?

$6^2$ (correct answer)
$6^6$
$6^8$
$3^2$
$12^2$

Explanation: When you see fractions with the same base raised to different powers, you're working with the quotient rule for exponents. This rule states that when dividing powers with the same base, you subtract the exponents:

\frac{a^m}{a^n} = a^{m-n}

. Let's apply this to

\frac{6^4}{6^2}

. Since both the numerator and denominator have base 6, we subtract the exponents:

6^{4-2} = 6^2

. You can verify this by thinking about what the original expression means:

\frac{6 \times 6 \times 6 \times 6}{6 \times 6}

. The two 6's in the denominator cancel with two of the 6's in the numerator, leaving

6 \times 6 = 6^2

. Choice A (

6^2

) is correct because it equals our simplified result. Choice B (

6^6

) represents a common error where students add the exponents instead of subtracting them. This would be the result if you were multiplying

6^4 \times 6^2

, not dividing. Choice C (

6^8

) occurs when students multiply the exponents, which would apply to a situation like

(6^4)^2

, not division of powers. Choice D (

3^2

) might tempt students who incorrectly think they should divide the base along with applying an exponent rule, but the quotient rule only affects the exponents when the bases are identical. Remember: when dividing powers with the same base, subtract the exponents. When multiplying, add them. When raising a power to a power, multiply the exponents.

Question 5

If $2^x = 32$ and $3^y = 27$ , what is the value of $x^y$ ?

25
125 (correct answer)
243
15
81

Explanation: This question tests your ability to solve exponential equations and then evaluate expressions with those solutions. When you see equations like

2^x = 32

and

3^y = 27

, you need to find the values of the variables by recognizing powers of the bases. To solve

2^x = 32

, think about what power of 2 equals 32. Since

2^5 = 32

, we have

x = 5

. Similarly, for

3^y = 27

, we need the power of 3 that equals 27. Since

3^3 = 27

, we have

y = 3

. Now we can find

x^y = 5^3 = 5 \times 5 \times 5 = 125

. Looking at the wrong answers: Choice (A) gives 25, which you'd get if you calculated

5^2

instead of

5^3

- this means you found

x

correctly but used

y = 2

instead of

y = 3

. Choice (C) gives 243, which equals

3^5

- this suggests you switched the values and calculated

y^x

instead of

x^y

. Choice (D) gives 15, which is simply

x \times y = 5 \times 3

- this means you added the exponents instead of using one as the base and the other as the exponent. When solving exponential equations on the SSAT, first convert the right side to a power of the base if possible, then read off the exponent. Always double-check which variable should be the base and which should be the exponent in your final calculation.

Question 6

Which of the following equals $2^{10} - 2^9$ ?

$2^8$
$2^9$ (correct answer)
$2^{10}$
$2^1$
$2^{19}$

Explanation: When you encounter expressions with the same base but different exponents being subtracted, look for ways to factor out common terms rather than calculating each power separately. To solve

2^{10} - 2^9

, notice that both terms share a factor of

2^9

. You can factor this out:

2^{10} - 2^9 = 2^9 \cdot 2^1 - 2^9 \cdot 1 = 2^9(2 - 1) = 2^9 \cdot 1 = 2^9

. Alternatively, you can think of this as

2^{10} - 2^9 = 2 \cdot 2^9 - 2^9 = 2^9(2 - 1) = 2^9

. Either way, the answer is

2^9

, which is choice B. Let's examine why the other choices are incorrect. Choice A (

2^8

) would be the result if you mistakenly thought

2^{10} - 2^9 = 2^{10-9} = 2^1 = 2

, then confused this with

2^8

. Choice C (

2^{10}

) ignores the subtraction entirely. Choice D (

2^1

) comes from the common error of subtracting exponents:

10 - 9 = 1

, so

2^1

. However, you cannot simply subtract exponents when subtracting exponential expressions. The key strategy here is recognizing factoring opportunities with exponential expressions. When you see terms with the same base, factor out the greatest common power before performing operations. This approach is much more efficient than calculating

2^{10} = 1024

and

2^9 = 512

, then subtracting to get 512. Remember: factor first, calculate last.

Question 7

If $3^a = 9$ and $3^b = 27$ , what is $3^{a+b}$ ?

36
81
243 (correct answer)
729
2187

Explanation: This question tests your understanding of exponent rules, specifically how to work with powers that have the same base. When you see exponential equations like these, think about converting the numbers to the same base first. Start by expressing 9 and 27 as powers of 3. Since

9 = 3^2

and

27 = 3^3

, you can rewrite the given equations as

3^a = 3^2

and

3^b = 3^3

. This means

a = 2

and

b = 3

. Now you can find

3^{a+b} = 3^{2+3} = 3^5

. To calculate

3^5

, multiply:

3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243

. The answer is C. Let's examine why the other choices are wrong. Choice A (36) might tempt you if you mistakenly added the original values:

9 + 27 = 36

. However, when dealing with exponents, you don't simply add the results. Choice B (81) equals

3^4

, which you might get if you miscalculated

a + b

as 4 instead of 5. Choice D (729) equals

3^6

, which could result from incorrectly thinking

a + b = 6

. Remember this key exponent rule: when multiplying powers with the same base, you add the exponents, so

3^a \times 3^b = 3^{a+b}

. For SSAT exponent problems, always convert to the same base when possible, then use the fundamental rules of exponents rather than trying to work with the calculated values directly.

Question 8

What is the value of $\frac{5^7}{5^4} \cdot 5^2$ ?

$5^3$
$5^5$ (correct answer)
$5^9$
$25^3$
$125^2$

Explanation: When you see expressions with exponents being multiplied or divided, you're working with the laws of exponents. These rules help you simplify complex expressions efficiently. Let's work through this step by step. You have

\frac{5^7}{5^4} \cdot 5^2

. First, handle the division: when dividing powers with the same base, subtract the exponents. So

\frac{5^7}{5^4} = 5^{7-4} = 5^3

. Now you have

5^3 \cdot 5^2

. When multiplying powers with the same base, add the exponents:

5^3 \cdot 5^2 = 5^{3+2} = 5^5

. This confirms that choice B is correct. Let's examine why the other answers are wrong. Choice A (

5^3

) represents only the first step of the calculation—you'd get this if you forgot to multiply by

5^2

at the end. Choice C (

5^9

) comes from a common mistake: adding all the exponents without properly handling the division first (

7 + 4 + 2 = 13

is wrong, but

7 + 2 = 9

suggests confusion about the division step). Choice D (

25^3

) might tempt you because

25 = 5^2

, but this completely misapplies the exponent rules and doesn't follow from the given expression. Remember this key strategy: always work left to right with exponent expressions, applying one rule at a time. Division means subtract exponents, multiplication means add exponents—but only when the bases are the same. Master these two rules and you'll handle most exponent problems confidently.

Question 9

Which of the following is equal to $\frac{16^3}{4^4}$ ?

$2^2$
$2^4$ (correct answer)
$4^2$
$4^{-1}$
$16^{-1}$

Explanation: When you encounter expressions with different bases that are powers of the same number, your key strategy is to rewrite everything using a common base. Both 16 and 4 are powers of 2, so let's express the entire fraction in terms of base 2. First, convert each term:

16 = 2^4

and

4 = 2^2

. Now substitute these into the original expression:

\frac{16^3}{4^4} = \frac{(2^4)^3}{(2^2)^4}

Using the power rule

(a^m)^n = a^{mn}

, this becomes:

\frac{2^{12}}{2^8}

When dividing powers with the same base, subtract the exponents:

2^{12-8} = 2^4

This matches answer choice B. Looking at the wrong answers: Choice A gives

2^2 = 4

, which would result from incorrectly calculating

2^{12-8}

2^2

. Choice C gives

4^2 = 16

, which you might get if you mistakenly simplified to

\frac{16}{4} = 4

and then squared it. Choice D gives

4^{-1} = \frac{1}{4}

, which could result from confusing the order of subtraction in the exponents or misapplying exponent rules. Study tip: Always convert to the smallest common base when working with exponents. Write out each step clearly: convert bases, apply power rules, then use division rules for exponents. This systematic approach prevents the calculation errors that create most wrong answer choices on these problems.

Question 10

What is the value of $3^4 \cdot 9^2$ ?

$3^6$
$3^8$ (correct answer)
$27^4$
$81^2$
$729$

Explanation: When you encounter expressions with exponents that need to be multiplied, the key is to express everything in terms of the same base so you can apply the rules of exponents effectively. Start by recognizing that

9 = 3^2

, so you can rewrite

9^2

(3^2)^2

. Using the power rule for exponents,

(3^2)^2 = 3^{2 \cdot 2} = 3^4

. Now your expression becomes

3^4 \cdot 3^4

. When multiplying powers with the same base, you add the exponents:

3^4 \cdot 3^4 = 3^{4+4} = 3^8

. Looking at the wrong answers: Choice A gives

3^6

, which would result from incorrectly adding the original exponents

4 + 2 = 6

without first converting

9^2

to base 3. Choice C,

27^4

, might tempt you if you mistakenly thought

3^4 = 27

(but

3^4 = 81

, and

3^3 = 27

). Choice D,

81^2

, could arise from correctly calculating

3^4 = 81

but then incorrectly thinking

81 \cdot 9^2 = 81^2

instead of properly handling the exponents. The correct answer is B:

3^8

. Strategy tip: When multiplying expressions with exponents, always convert to a common base first. Look for perfect powers (like

9 = 3^2

16 = 2^4

25 = 5^2

) and rewrite them before applying exponent rules. This approach will help you avoid calculation errors and see the correct path forward.

Question 11

Which expression equals $\frac{2^8 \cdot 4^3}{8^2}$ ?

$2^8$ (correct answer)
$2^{10}$
$2^{12}$
$4^6$
$8^4$

Explanation: When you encounter expressions with different bases that are all powers of the same number, your strategy should be to rewrite everything using a common base. Here, notice that 4 and 8 are both powers of 2, so convert everything to base 2. Start by rewriting each term:

4^3 = (2^2)^3 = 2^6

and

8^2 = (2^3)^2 = 2^6

. This gives you

\frac{2^8 \cdot 2^6}{2^6}

. Using exponent rules, when multiplying powers with the same base, you add exponents:

2^8 \cdot 2^6 = 2^{14}

. When dividing powers with the same base, you subtract exponents:

\frac{2^{14}}{2^6} = 2^{14-6} = 2^8

. Now let's examine why each answer choice is incorrect. Choice B (

2^{10}

) likely comes from incorrectly calculating

2^8 \cdot 4^3

2^8 \cdot 2^2 = 2^{10}

while forgetting to convert

4^3

properly or ignoring the denominator entirely. Choice C (

2^{12}

) results from converting

4^3

2^6

correctly but then making an error with the denominator, perhaps treating

8^2

2^2

instead of

2^6

. Choice D (

4^6

) might tempt you if you try to work directly with base 4, but this approach leads to complications since not all terms convert cleanly to base 4. Remember: when dealing with mixed bases in exponent problems, always convert to the smallest common base first. This eliminates confusion and makes the arithmetic straightforward using basic exponent rules.

Question 12

What is the value of $\frac{3^6}{3^2} \cdot \frac{3^3}{3^5}$ ?

$3^{-2}$
$3^2$ (correct answer)
$3^6$
$\frac{1}{9}$
$9$

Explanation: When you encounter expressions with exponents that need to be multiplied or divided, you're working with the laws of exponents. The key rules here are: when dividing powers with the same base, subtract the exponents (

\frac{a^m}{a^n} = a^{m-n}

), and when multiplying powers with the same base, add the exponents (

a^m \cdot a^n = a^{m+n}

). Let's work through this step by step. First, simplify each fraction separately:

$\frac{3^6}{3^2} = 3^{6-2} = 3^4$
$\frac{3^3}{3^5} = 3^{3-5} = 3^{-2}$

Now multiply these results:

3^4 \cdot 3^{-2} = 3^{4+(-2)} = 3^2

Looking at the wrong answers: Choice A gives

3^{-2}

, which is what you'd get if you only calculated the second fraction and forgot about the first one. Choice C gives

3^6

, which might result from incorrectly adding all the exponents without considering the division operations. Choice D gives

\frac{1}{9}

, which equals

3^{-2}

, so it's the same error as choice A but written in fraction form rather than exponential form. The correct answer is B)

3^2

. Study tip: When working with complex expressions involving exponents, break them down into smaller parts first, then combine. Always remember that division means subtracting exponents, and multiplication means adding them. Double-check your work by verifying that negative exponents make sense in context.

Question 13

What is $\frac{(5^2)^3}{5^4}$ ?

5
25 (correct answer)
125
625
3125

Explanation: This question tests your understanding of exponent rules, specifically how to simplify expressions with powers and division. When you see nested exponents like

(5^2)^3

, you need to apply the power rule: when raising a power to another power, multiply the exponents. So

(5^2)^3 = 5^{2 \times 3} = 5^6

. Now your expression becomes

\frac{5^6}{5^4}

. When dividing powers with the same base, subtract the exponents using the quotient rule:

5^6 \div 5^4 = 5^{6-4} = 5^2 = 25

. Let's examine why the other answers are incorrect. Choice (A) gives you 5, which equals

5^1

. You might get this if you incorrectly calculated

5^{6-4}

5^1

instead of

5^2

. Choice (C) gives you 125, which equals

5^3

. This could happen if you made an error in your exponent subtraction, perhaps calculating

6-4

as 3. Choice (D) gives you 625, which equals

5^4

. You might arrive at this by forgetting to subtract the bottom exponent entirely, leaving you with just the denominator's value. Remember this key strategy: with exponent problems, work step-by-step through the rules. First handle any nested exponents by multiplying, then deal with division by subtracting exponents. Writing out each step prevents careless arithmetic errors that lead to the wrong answer choices.

Question 14

If $3^{m+1} = 27^2$ , what is the value of $m$ ?

4
5 (correct answer)
6
8
9

Explanation: This question tests your ability to work with exponents and express numbers in different bases. When you see an equation with different bases that can be related, look for ways to express both sides using the same base. Start by recognizing that both 3 and 27 are powers of 3, since

27 = 3^3

. This means you can rewrite the right side of the equation using base 3. Since

27^2 = (3^3)^2

, you can use the power rule for exponents:

(a^m)^n = a^{mn}

. Therefore,

27^2 = (3^3)^2 = 3^{3 \times 2} = 3^6

. Now your equation becomes

3^{m+1} = 3^6

. When you have equal bases, the exponents must be equal, so

m+1 = 6

. Solving for

m

m = 6-1 = 5

. Let's check why the other answers don't work. Choice (A) gives

m = 4

, which would make

3^{4+1} = 3^5 = 243

, but

27^2 = 729

. Choice (C) gives

m = 6

, making

3^{6+1} = 3^7 = 2187

, which is too large. Choice (D) gives

m = 8

, making

3^{8+1} = 3^9

, which is far too large. The answer is (B) 5. Study tip: When working with exponential equations, always look for opportunities to express both sides using the same base. Remember common powers like

27 = 3^3

8 = 2^3

, and

16 = 2^4

to quickly identify these relationships.

Question 15

What is $\frac{10^5}{10^2} \div 10^1$ ?

$10^1$
$10^2$ (correct answer)
$10^3$
$10^6$
$10^8$

Explanation: When you see exponents being divided, you're working with the laws of exponents. This problem involves two key rules: dividing powers with the same base, and the order of operations. First, let's handle

\frac{10^5}{10^2}

. When dividing powers with the same base, you subtract the exponents:

10^5 \div 10^2 = 10^{5-2} = 10^3

. Now you have

10^3 \div 10^1

. Applying the same rule:

10^3 \div 10^1 = 10^{3-1} = 10^2

. Let's see why the wrong answers occur. Choice A (

10^1

) happens if you mistakenly subtract all three exponents from left to right:

5 - 2 - 1 = 2

, but then somehow get

10^1

. Choice C (

10^3

) results from only doing the first division and forgetting about the second

\div 10^1

. Choice D (

10^6

) comes from a major error—either adding exponents instead of subtracting (

5 + 2 - 1 = 6

) or misremembering the division rule entirely. The correct answer is B:

10^2

. Remember this key strategy: when you see multiple operations with exponents, work step by step from left to right, and always subtract exponents when dividing powers with the same base. Double-check by asking yourself if your final answer makes sense—since we're dividing by larger and larger numbers, our result should be getting smaller, which

10^5

10^2

correctly shows.

Question 16

If $6^a = 36$ and $6^b = 216$ , what is $6^{a+b}$ ?

252
1296
7776 (correct answer)
46656
279936

Explanation: When you encounter exponential equations like this, you're working with the fundamental properties of exponents. The key insight is that when you multiply powers with the same base, you add the exponents:

a^m \cdot a^n = a^{m+n}

. First, let's find the values of

a

and

b

. Since

6^a = 36

, you need to determine what power of 6 gives you 36. Notice that

6^2 = 36

, so

a = 2

. Similarly, for

6^b = 216

, you need

6^3 = 216

, so

b = 3

. Now you can find

6^{a+b} = 6^{2+3} = 6^5

. Calculate this:

6^5 = 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 7776

. This confirms answer choice C is correct. Let's examine why the other answers are wrong. Choice A (252) is far too small—it's less than even

6^3

. Choice B (1296) equals

6^4

, which you'd get if you mistakenly calculated

a + b = 4

instead of 5. Choice D (46656) equals

6^6

, which you might get by incorrectly adding

a + b = 6

. The most common error here is miscalculating the individual exponents. Practice recognizing perfect powers: memorize that

6^2 = 36

6^3 = 216

6^4 = 1296

, etc. When you see exponential problems on the SSAT, always double-check your exponent arithmetic—small mistakes in finding

a

and

b

will lead you directly to trap answers.

Question 17

Which expression is equivalent to $\frac{8^4}{2^6}$ ?

$4^2$
$2^6$ (correct answer)
$4^4$
$2^8$
$2^{10}$

Explanation: When you encounter expressions with exponents that need simplification, the key is to express everything using the same base whenever possible. This allows you to use exponent rules effectively. To solve

\frac{8^4}{2^6}

, first recognize that 8 can be written as a power of 2:

8 = 2^3

. This means

8^4 = (2^3)^4

. Using the power rule

(a^m)^n = a^{mn}

, we get

(2^3)^4 = 2^{12}

. Now the expression becomes

\frac{2^{12}}{2^6}

. Using the quotient rule

\frac{a^m}{a^n} = a^{m-n}

, we get

2^{12-6} = 2^6

. Looking at the wrong answers: Choice (A)

4^2 = 16

is much smaller than our result, since

2^6 = 64

. Choice (C)

4^4 = 256

is too large compared to

2^6 = 64

. Choice (D)

2^8 = 256

is also too large—this would be the result if you incorrectly added the exponents instead of subtracting them. The correct answer is (B)

2^6

. Strategy tip: When simplifying expressions with different bases, always check if you can rewrite them using a common base (often 2, 3, or 10). This transforms complex-looking problems into straightforward applications of exponent rules. Also, remember that division means you subtract exponents when the bases are the same.

Question 18

What is the value of $\frac{4^5 \cdot 2^3}{8^2}$ ?

32
64
128 (correct answer)
256
512

Explanation: When you encounter expressions with multiple bases and exponents, the key is to express everything using the same base so you can apply exponent rules effectively. Let's rewrite each term using base 2:

4^5 = (2^2)^5 = 2^{10}

2^3

stays as

2^3

, and

8^2 = (2^3)^2 = 2^6

. Now our expression becomes:

\frac{2^{10} \cdot 2^3}{2^6}

Using the multiplication rule for exponents (when bases are the same, add exponents), the numerator becomes

2^{10+3} = 2^{13}

. So we have

\frac{2^{13}}{2^6}

. Using the division rule for exponents (when bases are the same, subtract exponents), this equals

2^{13-6} = 2^7 = 128

. Answer choice (A) 32 represents

2^5

, which you might get if you incorrectly calculated

4^5

2^5

instead of

2^{10}

. Answer choice (B) 64 equals

2^6

, which could result from forgetting to multiply by

2^3

in the numerator. Answer choice (D) 256 equals

2^8

, which might occur if you made an error when subtracting exponents in the final step. The correct answer is (C) 128. Study tip: Always convert to a common base when working with exponents involving numbers like 2, 4, and 8. Remember that

4 = 2^2

and

8 = 2^3

, so you can rewrite everything in terms of base 2 to simplify calculations.

Question 19

If $3^{2x} = 81$ and $2^y = 8$ , what is $x + y$ ?

5 (correct answer)
6
7
8
9

Explanation: When you encounter exponential equations like these, you're working with the concept that if two powers with the same base are equal, their exponents must be equal. The key is rewriting both sides using the same base. For the first equation,

3^{2x} = 81

, you need to express 81 as a power of 3. Since

81 = 3^4

, you can rewrite the equation as

3^{2x} = 3^4

. When the bases are equal, the exponents must be equal, so

2x = 4

, which means

x = 2

. For the second equation,

2^y = 8

, express 8 as a power of 2. Since

8 = 2^3

, you have

2^y = 2^3

, so

y = 3

. Therefore,

x + y = 2 + 3 = 5

. Looking at the wrong answers: Choice B (6) might result from incorrectly finding

x = 3

and

y = 3

, perhaps by confusing

3^4 = 81

with

3^3

. Choice C (7) could come from finding

x = 4

and

y = 3

, which happens if you mistakenly set

2x = 4

and conclude

x = 4

instead of dividing by 2. Choice D (8) might result from setting

x = 4

and

y = 4

, possibly from misremembering that

2^4 = 16

, not 8. The key strategy is to always convert to the same base on both sides of exponential equations. Memorize common powers (like

2^3 = 8

3^4 = 81

) to solve these quickly and avoid calculation errors.

Question 20

If $5^{x+2} = 625$ , what is $5^{x-1}$ ?

5 (correct answer)
25
125
625
3125

Explanation: This problem tests your ability to work with exponential equations and use the properties of exponents to find related expressions. To solve

5^{x+2} = 625

, you first need to express 625 as a power of 5. Since

5^1 = 5

5^2 = 25

5^3 = 125

, and

5^4 = 625

, you can rewrite the equation as

5^{x+2} = 5^4

. When the bases are equal, the exponents must be equal, so

x+2 = 4

, which means

x = 2

. Now you can find

5^{x-1} = 5^{2-1} = 5^1 = 5

. Looking at the wrong answers: Choice B (25) equals

5^2

, which you'd get if you mistakenly thought

x = 3

or if you calculated

5^{x-1}

5^{3-1}

. Choice C (125) equals

5^3

, which you might choose if you found

x = 2

correctly but then calculated

5^{x+1}

instead of

5^{x-1}

. Choice D (625) equals

5^4

, which is the original value from the given equation—a common trap for students who confuse what they're solving for. When working with exponential equations, always convert everything to the same base when possible, then set the exponents equal. Double-check that you're calculating the expression that's actually being asked for, not just finding the value of the variable.