SSAT Upper Level

Exponents & Roots

Q: What is a fractional exponent?

A fractional exponent like `x^{\frac{1}{2}}` is just a secret disguise for a square root `\sqrt{x}`! And `x^{\frac{1}{3}}` is a cube root. Just think of the bottom number of the fraction as the type of root.

Integer and fractional exponents, square roots, and scientific notation — excludes variable exponent manipulation (see radical-rational-expressions) and polynomial expressions (see polynomials-quadratics)

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Imagine a magic spell that makes your allowance double every day! That's what exponents do—they are math's ultimate power-up. 🍄✨ An exponent is just a tiny number floating at the top right of a normal number. It tells the big number how many times to multiply by itself. So, $3^{2}$ isn't $3 \times 2$ , it's $3 \times 3$ ! Think of it like a video game clone glitch. If you have 3 ninja clones, and each of them makes 3 clones, suddenly you have 9 ninjas!

Roots are the exact opposite. If an exponent is building a giant LEGO tower, a root is breaking it back down to its base pieces. 🧱 A square root asks, 'What number multiplied by itself gives me this big number?' So, the square root of 25 is 5, because 5 times 5 is 25.

On the SSAT, the test-makers will try to trick you with negative exponents, fractions, and scientific notation (which is just a fancy way scientists write super huge or super tiny numbers). But don't worry! Once you know the secret rules of the power-up mushroom, you'll crush these questions easily! 🚀 Just remember to take it step by step, use your estimation skills, and never let a tiny floating number intimidate you!

Key Formula

The Flip-It Rule for negative exponents:

x^{- n} = \frac{1}{x ^{n}}

. If you see a negative exponent on a fraction, just flip the fraction upside down to make the exponent happy and positive! Like this:

(\frac{a}{b})^{- n} = (\frac{b}{a})^{n}

Practice Questions

4 practice questions for SSAT Upper Level

Q1 Hard

Which of the following is equivalent to

(5 \times 1 0^{4}) \times (8 \times 1 0^{6})

4.0 \times 1 0^{10}

4.0 \times 1 0^{11}

4.0 \times 1 0^{24}

4.0 \times 1 0^{25}

40 \times 1 0^{24}

Show Solution

First, multiply the numbers outside the powers of 10: $5 \times 8 = 40$ . Next, multiply the powers of 10 by adding their exponents: $1 0^{4} \times 1 0^{6} = 1 0^{4 + 6} = 1 0^{10}$ . This gives $40 \times 1 0^{10}$ . To convert this to standard scientific notation, move the decimal point one place to the left, which increases the exponent by 1: $4.0 \times 1 0^{11}$ .

Answer: B

Q2 Hard

Which of the following is equivalent to

48 - 12

23

32

36

43

E 6

Show Solution

First, simplify each square root by finding the largest perfect square factor. For the first term, $48 = 16 \times 3 = 43$ . For the second term, $12 = 4 \times 3 = 23$ . Substitute these simplified terms back into the expression: $43 - 23 = 23$ .

Answer: A

Q3 Hard

The square of 7 is divided by the cube root of 125. What is the remainder?

A 0

B 1

C 2

D 3

E 4

Show Solution

First, find the square of 7, which is $7^{2} = 49$ . Next, find the cube root of 125, which is 5 (since $5 \times 5 \times 5 = 125$ ). Finally, divide 49 by 5. $49 \div 5 = 9$ with a remainder of 4.

Answer: E

Q4 Hard

Which of the following is equivalent to

\frac{3 ^{6} \times 3 ^{4}}{3 ^{7}}

A 3

B 9

C 27

D 81

E 243

Show Solution

When multiplying terms with the same base, add the exponents: $3^{6} \times 3^{4} = 3^{6 + 4} = 3^{10}$ . The expression then becomes $\frac{3 ^{10}}{3 ^{7}}$ . When dividing terms with the same base, subtract the exponents: $3^{10 - 7} = 3^{3}$ . Finally, evaluate the expression: $3^{3} = 3 \times 3 \times 3 = 27$ .

Answer: C

Tips & Strategies

When dealing with decimals inside square roots, turn them into fractions first! $0.25$ looks scary, but $\frac{25}{100}$ is easy to solve as $\frac{5}{10}$ .
For scientific notation division or multiplication, treat the regular numbers and the $10$ s like two separate teams. Solve the regular numbers first, then use your exponent rules on the $10$ s!
When you see a fractional exponent like $x^{\frac{1}{2}}$ , rewrite it immediately as a square root $x$ on your scratch paper so your brain recognizes it.

Common Mistakes

Watch out for multiplying only twice! A huge mistake is thinking $3^{3}$ means $3 \times 3 = 9$ . The exponent 3 means three copies: $3^{3} = 3 \times 3 \times 3 = 27$ !
Don't forget that negative exponents do NOT make the number negative! $2^{- 3}$ is not $- 8$ . It is $\frac{1}{2 ^{3}}$ , which equals $\frac{1}{8}$ .

Frequently Asked Questions

What does an exponent of 0 mean?

Any non-zero number to the power of 0 is exactly 1! Even a million to the power of 0 ( $100000 0^{0}$ ) is just 1. It's a fun math trick!

Will I have to calculate giant roots on the SSAT?

Nope! The SSAT tests your logic, not your ability to act like a calculator. If you see a big root, look for a way to estimate it or simplify it using perfect squares you already know.

What is a fractional exponent?

A fractional exponent like $x^{\frac{1}{2}}$ is just a secret disguise for a square root $x$ ! And $x^{\frac{1}{3}}$ is a cube root. Just think of the bottom number of the fraction as the type of root.

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