ISEE Upper Level

Exponents & Roots

Q: Do I need to know fractional exponents for the lower level ISEE?

Usually not! Fractional exponents like `x^{\frac{1}{2}}` mostly show up on the Upper Level ISEE. For the Middle Level, focus on basic squares and square roots.

Q: What do I do if I forget the exponent rules during the test?

Write it out! If you forget what `\frac{x^5}{x^3}` is, just write `\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x}` and cross out the matching pairs on top and bottom. You'll be left with `x^2`.

Q: What is any number to the power of zero?

It is always 1! Whether it's `5^0` or `1,000,000^0`, the answer is 1. It's a weird math rule, but it's super handy to remember for the test!

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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Have you ever wished for a cloning machine to help you finish your chores? 🤖 In math, exponents are exactly that—a super-fast cloning machine for numbers! When you see a number with a tiny number floating above it, like $3^{4}$ , it means the big number (the base) is cloning itself and multiplying. So, $3^{4}$ means $3 \times 3 \times 3 \times 3$ . It’s way faster than writing it all out!

Roots, like the square root $25$ , are the exact opposite. They are the 'un-cloning' machines. They ask, 'What number multiplied by itself gives me 25?' Since $5 \times 5 = 25$ , the answer is 5! 🍕 Think of it like a pizza: exponents make the pizza huge, and roots shrink it back down to the original dough.

On the ISEE, you'll see exponents and roots in both the Quantitative Reasoning section (where you compare two columns) and the Mathematics Achievement section (where you solve equations). The test makers love to see if you know the secret rules, like what happens when an exponent is negative or a fraction. Don't worry, once you learn the rules, it's like having a cheat code for a video game! 🎮 Remember, there is no penalty for guessing on the ISEE, so always take your best shot!

Key Formula

When you see a negative exponent, it means to flip the base into a fraction!

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

. If the base is already a fraction, just flip it upside down:

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

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

What is the value of the numerical expression

\frac{5.4 \times 1 0 ^{6}}{9.0 \times 1 0 ^{- 3}}

6.0 \times 1 0^{8}

6.0 \times 1 0^{9}

6.0 \times 1 0^{2}

6.0 \times 1 0^{3}

Show Solution

Divide the coefficients: $\frac{5.4}{9.0} = 0.6$ . Divide the powers of 10: $\frac{1 0 ^{6}}{1 0 ^{- 3}} = 1 0^{6 - (- 3)} = 1 0^{9}$ . So the product is $0.6 \times 1 0^{9}$ . Converting to proper scientific notation: $0.6 \times 1 0^{9} = 6.0 \times 1 0^{8}$ .

Answer: A

Q2 Hard

Compare the two quantities.

Column A

(\frac{1}{5})^{- 3}

Column B

(\frac{1}{5})^{- \frac{1}{2}}

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

Column A: $(\frac{1}{5})^{- 3} = 5^{3} = 125$ . Column B: $(\frac{1}{5})^{- \frac{1}{2}} = 5$ , which is between 2 and 3. Therefore Column A (125) is much greater than Column B ( $5 \approx 2.24$ ).

Answer: A

Q3 Hard

What is the value of the expression

\frac{2 ( 2 ^{3} + 2 ^{4} )}{8 ( 2 + 4 )}

\frac{1}{2}

B 1

C 2

D 4

Show Solution

Evaluate the numerator first: $2^{3} = 8$ and $2^{4} = 16$ , so $2^{3} + 2^{4} = 24$ . Multiplying by 2: numerator $= 2 \times 24 = 48$ . Evaluate the denominator: $2 + 4 = 6$ , so denominator $= 8 \times 6 = 48$ . The fraction equals $\frac{48}{48} = 1$ .

Answer: B

Q4 Hard

Compare the two quantities, where

n

is a positive integer greater than 1.

Column A

n^{3}

Column B

n^{4}

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

Since $n$ is a positive integer greater than 1, we can compare $n^{3}$ and $n^{4}$ by dividing both by $n^{3}$ (which is positive): this gives $1$ versus $n$ . Because $n > 1$ , Column B ( $n^{4} = n^{3} \times n$ ) is always greater than Column A ( $n^{3}$ ). For example, if $n = 2$ : Column A is $8$ , Column B is $16$ . If $n = 3$ : Column A is $27$ , Column B is $81$ . Column B is always greater.

Answer: B

Tips & Strategies

💡 For Quantitative Comparison questions, don't calculate everything perfectly if you don't have to! If you know $1 7^{2}$ is a big whole number and $1 7^{\frac{1}{2}}$ is a small square root, you already know which is bigger.
⏱️ Memorize your perfect squares up to $1 5^{2} = 225$ and your powers of 2 up to $2^{6} = 64$ . It will save you tons of time on the ISEE Mathematics Achievement section!
🧠 When multiplying numbers with the same base, ADD the exponents. When dividing them, SUBTRACT the exponents. Think: 'Multiply = More (Add), Divide = Less (Subtract)'.

Common Mistakes

⚠️ Watch out for negative exponents! They do NOT make the number negative. They just flip the number into a fraction. $3^{- 2}$ is $\frac{1}{9}$ , not $- 9$ .
⚠️ Don't forget the difference between $2^{3}$ and $2 \times 3$ . $2^{3}$ is $2 \times 2 \times 2 = 8$ , but $2 \times 3 = 6$ . Exponents mean multiplying the number by itself!

Frequently Asked Questions

Do I need to know fractional exponents for the lower level ISEE?

Usually not! Fractional exponents like $x^{\frac{1}{2}}$ mostly show up on the Upper Level ISEE. For the Middle Level, focus on basic squares and square roots.

What do I do if I forget the exponent rules during the test?

Write it out! If you forget what $\frac{x ^{5}}{x ^{3}}$ is, just write $\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x}$ and cross out the matching pairs on top and bottom. You'll be left with $x^{2}$ .

Is there a penalty if I guess wrong on an exponent question?

Nope! The ISEE does not have a wrong-answer penalty. If you are totally stuck, try to eliminate one or two crazy answers, and then pick your favorite letter. Never leave a bubble blank!

What is any number to the power of zero?

It is always 1! Whether it's $5^{0}$ or $1, 000, 00 0^{0}$ , the answer is 1. It's a weird math rule, but it's super handy to remember for the test!

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