How to find the properties of an exponent

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SSAT Upper Level Quantitative › How to find the properties of an exponent

Questions 1 - 10

Evaluate: $\dpi{100} (3^{3})^{2}$

$\dpi{100} 3^{6}$

$\dpi{100} 3^{5}$

$\dpi{100} 243$

$\dpi{100} 729$

Explanation

A power raised to a power indicates that you multiply the two powers.

$\dpi{100} (3^{3})^{2}=3^{3\cdot 2}=3^{6}$

and are both positive integers; A is odd. What can you say about the number

$(A + B + 1)^{A+B+1}$ ?

$(A + B + 1)^{A+B+1}$ is odd if is odd, and even if is even.

$(A + B + 1)^{A+B+1}$ is even if is odd, and odd if is even.

$(A + B + 1)^{A+B+1}$ is odd if is odd, and can be odd or even if is even.

$(A + B + 1)^{A+B+1}$ is even if is odd, and can be odd or even if is even.

$(A + B + 1)^{A+B+1}$ is even if is even, and can be odd or even if is odd.

Explanation

If is odd, then , the sum of three odd integers, is odd; an odd number taken to any positive integer power is odd.

If is even, then , the sum of two odd integers and an even integer, is even; an even number taken to any positive integer power is even.

Therefore, $(A + B + 1)^{A+B+1}$ always assumes the same odd/even parity as .

$\dpi{100} (0.75)^{2}=$

$\dpi{100} \frac{9}{16}$

$\dpi{100} 49.5$

$\dpi{100} 58.5$

$\dpi{100} 5.85$

Explanation

Convert .75 to a fraction. $\dpi{100} \frac{75}{100}=\frac{3}{4}$ .

Now multiply $\dpi{100} \frac{3}{4}\times \frac{3}{4}=\frac{3\times 3}{4\times 4}=\frac{9}{16}$

$\dpi{100} Evaluate: (0.50^{2})$

$\dpi{100} 0.25$

$\dpi{100} 1$

$\dpi{100} 25$

$\dpi{100} 2.5$

Explanation

We can either write $\dpi{100} 0.5\times 0.5$ , or we can convert this to a fraction and write

$\dpi{100} \frac{1}{2}\times \frac{1}{2}$

$\dpi{100} \frac{1}{2}\times \frac{1}{2}=\frac{1\times 1}{2\times 2}=\frac{1}{4}$

$\dpi{100} \frac{1}{4}$ in decimal form is 0.25.

$(-4)^{2} + (-4)^{3}$

Explanation

$(-4)^{2} = -4 \cdot (-4) = +(4\cdot 4) = 16$

$(-4)^{3} = -4 \cdot (-4)\cdot (-4) = 16 \cdot (-4) = - (16 \cdot 4) = -64$

Therefore,

$(-4)^{2} + (-4)^{3} = 16 + (-64) = - (64-16) = -48$

Simplify:

$\frac{8x ^{-2}}{(3x)^{2}}$

$\frac{8}{9x ^{4}}$

$\frac{8}{3x ^{4}}$

$\frac{8}{3}$

$\frac{8}{9}$

$\frac{8}{9x}$

Explanation

$\frac{8x ^{-2}}{(3x)^{2}}$

To solve this problem, we start with the parentheses and exponents in the denominator.

$= \frac{8x ^{-2}}{3^{2}x^{2}} = \frac{8x ^{-2}}{9x^{2}}$

Next, we can bring the from the denominator up to the numerator by making the exponent negative.

$= \frac{8x ^{-2-2}}{9} = \frac{8x ^{-4}}{9}$

Finally, to get rid of the negative exponent we can bring it back down to the denominator.

$= \frac{8}{9x ^{4}}$

Assume all variables to be nonzero.

Simplify: $6 (xyz)^{0} + 4(xyz)^{0}$

Explanation

Any nonzero expression to the power of zero is equal to 1:

$6 (xyz)^{0} + 4(xyz)^{0} = 6 \times 1 + 4 \times 1 = 6 + 4 = 10$

What is $13\times13\times13\times13$ in exponential notation?

$4^{13}$

$4\times4\times4$

$13\times4$

$4\times13$

Explanation

Exponential notation includes a base number and an exponent. The base number is the number that is being multiplied and the exponent is how many times the base number is being multiplied to itself.

In this case, is our base number and it's being multiplied to itself times, so that is our exponent.

Evaluate:

$\frac{\left ( -5\right ) ^{5} - 5^{5}}{\left ( -5\right ) ^{5} + 5^{5}}$

The quantity is undefined.

$-\frac{1}{6,250}$

$-\frac{2}{3,125}$

Explanation

$5^{5} = 5 \times 5 \times 5 \times 5 \times 5 = 3,125$

$\left (-5 \right ) ^{5} = -\left ( 5^{5} \right ) = -3,125$

$\frac{\left ( -5\right ) ^{5} - 5^{5}}{\left ( -5\right ) ^{5} + 5^{5}} = \frac{-3,125 -3,125}{-3,125 + 3,125} =\frac{-6,250}{0}$

This is an undefined quantity.

Express the result in scientific notation:

$(2 \times 10^{3}) (3 \times 10^{4}) (5 \times 10^{5})$

$3 \times 10^{13}$

$30 \times 10^{12}$

$3 \times 10^{12}$

$3 \times 10^{61}$

$30 \times 10^{60}$

Explanation

Apply the product of powers property:

$(2 \times 10^{3}) (3 \times 10^{4}) (5 \times 10^{5})$

$= 2 \times 3 \times 5 \times 10^{3} \times 10^{4} \times 10^{5}$

$= 30 \times 10^{3+4+5}$

$= 30 \times 10^{12}$

Since this is not in scientific notation, adjust as follows:

$30 \times 10^{12}$

$= 3 \times 10^{1} \times 10^{12}$

$= 3 \times 10^{1+12}$

$= 3 \times 10^{13}$

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