Exponents - SAT Math

Card 0 of 2288

Question

Which of the following is equal to 410 + 410 + 410 + 410 + 411?

Answer

We can start by rewriting 411 as 4 * 410. This will allow us to collect the like terms 410 into a single term.

410 + 410 + 410 + 410 + 411

= 410 + 410 + 410 + 410 + 4 * 410

= 8 * 410

Because the answer choices are written with a base of 2, we need to rewrite 8 and 4 using bases of two. Remember that 8 = 23, and 4 = 22.

8 * 410

= (23)(22)10

We also need to use the property of exponents that (ab)c = abc. We can rewrite (22)10 as 22x10 = 220.

(23)(22)10

= (23)(220)

Finally, we must use the property of exponents that ab * ac = ab+c.

(23)(220) = 223

The answer is 223.

Compare your answer with the correct one above

Question

If 3 + 3_n_+3 = 81, what is 3_n_+2 ?

Answer

3 + 3_n_+3 = 81

In this equation, there is a common factor of 3, which can be factored out.

Thus, 3(1 + 3_n_+2) = 81

Note: when 3 is factored out of 3_n_+3, the result is 3_n_+2 because (3_n_+3 = 31 * 3_n_+2). Remember that exponents are added when common bases are multiplied. Also remember that 3 = 31.

3(1 + 3_n_+2) = 81

(1 + 3_n_+2) = 27

3_n_+2 = 26

Note: do not solve for n individually. But rather seek to solve what the problem asks for, namely 3n+2.

Compare your answer with the correct one above

Question

Simplify: (x3 * 2x4 * 5y + 4y2 + 3y2)/y

Answer

Let's do each of these separately:

x3 * 2x4 * 5y = 2 * 5 * x3 * x4 * y = 10 * x7 * y = 10x7y

4y2 + 3y2 = 7y2

Now, rewrite what we have so far:

(10x7y + 7y2)/y

There are several options for reducing this. Remember that when we divide, we can "distribute" the denominator through to each member. That means we can rewrite this as:

(10x7y)/y + (7y2)/y

Subtract the y exponents values in each term to get:

10x7 + 7y

Compare your answer with the correct one above

Question

If $\sqrt{x}=y^{2}=6$ , then which of the following is equivalent to $x^3y^4$ ?

Answer

We can break up the equation into two smaller equations involving only x and y. Then, once we solve for x and y, we can find the value of $x^3y^4$ .

Compare your answer with the correct one above

Question

If $5^2 \times 5^n = 5^{12}$ , what is the value of ?

Answer

Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.

Compare your answer with the correct one above

Question

Simplify: y3x4(yx3 + y2x2 + y15 + x22)

Answer

When you multiply exponents, you add the common bases:

y4 x7 + y5x6 + y18x4 + y3x26

Compare your answer with the correct one above

Question

Solve for x.

23 + 2x+1 = 72

Answer

The answer is 5.

8 + 2x+1 = 72

2x+1 = 64

2x+1 = 26

x + 1 = 6

x = 5

Compare your answer with the correct one above

Question

Answer

Compare your answer with the correct one above

Question

If $3^{4x+6}=27^{2x}$ , what is the value of ?

Answer

Using exponents, 27 is equal to 33. So, the equation can be rewritten:

34_x_ + 6 = (33)2_x_

34_x_ + 6 = 36_x_

When both side of an equation have the same base, the exponents must be equal. Thus:

4_x_ + 6 = 6_x_

6 = 2_x_

x = 3

Compare your answer with the correct one above

Question

If _a_2 = 35 and _b_2 = 52 then _a_4 + _b_6 = ?

Answer

_a_4 = _a_2 * _a_2 and _b_6= _b_2 * _b_2 * _b_2

Therefore _a_4 + _b_6 = 35 * 35 + 52 * 52 * 52 = 1,225 + 140,608 = 141,833

Compare your answer with the correct one above

Question

If $9^{(x + 5)}+3^{2(x+5)}=162$ , what is the value of ?

Answer

Since we have two ’s in $9^{(x + 5)} + 3^{2(x + 5)}$ we will need to combine the two terms.

For $3^{2(x + 5) }$ this can be rewritten as

$(3^2)^{ (x + 5)} = 9^ {(x + 5)}$

So we have $9^{ (x + 5) }+ 9^{ (x + 5)} = 162$ .

Or $2 (9^{ (x + 5)}) = 162$

Divide this by : $9^{ (x + 5)} = 81 = 9^ 2$

Thus or

*Hint: If you are really unsure, you could have plugged in the numbers and found that the first choice worked in the equation.

Compare your answer with the correct one above

Question

54 / 25 =

Answer

25 = 5 * 5 = 52. Then 54 / 25 = 54 / 52.

Now we can subtract the exponents because the operation is division. 54 / 52 = 54 – 2 = 52 = 25. The answer is therefore 25.

Compare your answer with the correct one above

Question

What is the value of such that ?

Answer

We can solve by converting all terms to a base of two. 4, 16, and 32 can all be expressed in terms of 2 to a standard exponent value.

$4=2^2\rightarrow 4^2=(2^2)^2$

$16=2^4\rightarrow 16^3=(2^4)^3$

$32=2^5\rightarrow 32^2=(2^5)^2$

We can rewrite the original equation in these terms.

Simplify exponents.

$2^n=(2^4)(2^{12})(2^{10})$

Finally, combine terms.

$2^n=2^{4+12+10}=2^{26}$

From this equation, we can see that .

Compare your answer with the correct one above

Question

Which of the following is eqivalent to 5_b_ – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) , where b is a constant?

Answer

We want to simplify 5_b_ – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) .

Notice that we can collect the –5(b–1) terms, because they are like terms. There are 5 of them, so that means we can write –5(b–1) – 5(b–1) – 5(b–1) – 5(b–1) – 5(b–1) as (–5(b–1))5.

To summarize thus far:

5_b_ – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–1) = 5_b +(–5(_b–_1))5

It's important to interpret –5(b–1) as (–1)5(b–1) because the –1 is not raised to the (b – 1) power along with the five. This means we can rewrite the expression as follows:

5_b_ +(–5(b–1))5 = 5_b_ + (–1)(5(b–1))(5) = 5_b_ – (5(b–1))(5)

Notice that 5(b–1) and 5 both have a base of 5. This means we can apply the property of exponents which states that, in general, abac = a b+c. We can rewrite 5 as 51 and then apply this rule.

5_b_ – (5(_b–1))(5) = 5_b – (5(_b–1))(51) = 5_b – 5(_b–_1+1)

Now, we will simplify the exponent b – 1 + 1 and write it as simply b.

5_b_ – 5(b–1+1) = 5_b* – 5_b* = 0

The answer is 0.

Compare your answer with the correct one above

Question

If $\dpi{100} \small r$ and $\dpi{100} \small s$ are positive integers, and $\dpi{100} \small 25\left ( 5^{r} \right )=5^{s-2}$ , then what is $\dpi{100} \small s$ in terms of $\dpi{100} \small r$ ?

Answer

$\dpi{100} \small 25\left ( 5^{r} \right )$ is equal to $5^{2}\left ( 5^{r}\right )$ which is equal to $\dpi{100} \small \left ( 5^{r+2} \right )$ . If we compare this to the original equation we get $\dpi{100} \small r+2=s-2\rightarrow s=r+4$

Compare your answer with the correct one above

Question

Solve for $x$ :

$4^{4}+4^{4}+4^{4}+4^{4}=2^{x}$

Answer

Combining the powers, we get $1024=2^{x}$ .

From here we can use logarithms, or simply guess and check to get $x=10$ .

Compare your answer with the correct one above

Question

If $x^2=11$ , then what does $x^4$ equal?

Answer

$x^4=(x^2)(x^2)=11\cdot 11=121$

Compare your answer with the correct one above

Question

Simplify. All exponents must be positive.

$\left ( x^{-2}y^{3} \right )\left ( x^{5}y^{-4} \right )$

Answer

Step 1: $\left ( x^{-2}x^{5} \right )= x^{3}$

Step 2: $\left ( y^{3}y^{-4} \right )= y^{-1}= \frac{1}{y}$

Step 3: (Correct Answer): $\frac{x^{3}}{y}$

Compare your answer with the correct one above

Question

Simplify. All exponents must be positive.

$\frac{x^{-3}y^{5}}{x^{2}y^{-1}}$

Answer

Step 1: $\frac{y^{5}}{\left ( x^{3}x^{2} \right )\left \right )y^{-1}}$

Step 2: $\frac{\left ( y^{5}y^{1} \right )}{x^{3}x^{2}}$

Step 3: $\frac{y^{6}}{x^{5}}$

Compare your answer with the correct one above

Question

$\frac{\left ( -11 \right )^{-8}}{\left ( -11\right )^{12}}$

Answer must be with positive exponents only.

Answer

Step 1: $\frac{1}{\left ( -11 \right )^{12}\left ( -11 \right )^{8}}$

Step 2: The above is equal to $\frac{1}{\left ( -11 \right )^{20}}$

Compare your answer with the correct one above

Tap the card to reveal the answer