Skills

Exponent and Radical Rules

Same base: add exponents to multiply, subtract to divide, multiply exponents for a power of a power.

Practice 1: Procedural and Symbolic Fluency

Part 1 · Hook

Why this matters

A phone storage plan doubles every year: 2, 4, 8, 16 gigabytes. Writing that as a power of 2, and knowing what a negative or fractional power means, is the algebra behind every growth and decay model in Unit 2.

Part 2 · Prerequisite check

Quick check before you start

1. What is 343^4?

  1. 8181
  2. 1212
  3. 6464
  4. 2727
Show the answer

3⋅3⋅3⋅3=813 \cdot 3 \cdot 3 \cdot 3 = 81.

  • Correct: 8181:
  • 1212:
  • 6464:
  • 2727:

2. Expand (x2)(x3)(x^2)(x^3).

  1. x5x^5
  2. x6x^6
  3. 2x52x^5
  4. x23x^{23}
Show the answer

x2⋅x3=x⋅x⋅x⋅x⋅x=x5x^2 \cdot x^3 = x \cdot x \cdot x \cdot x \cdot x = x^5.

  • Correct: x5x^5:
  • x6x^6:
  • 2x52x^5:
  • x23x^{23}:

Part 3 · See it

See it first

Graphs of y = x^(1/2) and y = x^(1/3) for x from 0 to 9, meeting at (0, 0) and (1, 1); the square root passes (4, 2) and (9, 3), and the cube root passes (8, 2).
Rational exponents are roots: x1/2=xx^{1/2} = \sqrt{x} reaches 22 at x=4x = 4, and x1/3=x3x^{1/3} = \sqrt[3]{x} reaches 22 at x=8x = 8. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

  1. Multiply powers of the same baseadd the exponents: bmbn=bm+nb^m b^n = b^{m + n}
  2. Raise a power to a powermultiply the exponents: (bm)n=bmn(b^m)^n = b^{mn}
  3. Use a negative or fractional exponenttake a reciprocal (b−n=1bnb^{-n} = \frac{1}{b^n}) or a root (b1/n=bnb^{1/n} = \sqrt[n]{b})

Part 5 · Key ideas

Key ideas

  • bm⋅bn=bm+nb^m \cdot b^n = b^{m + n}, bmbn=bm−n\frac{b^m}{b^n} = b^{m - n}, (bm)n=bmn(b^m)^n = b^{mn}, (ab)n=anbn(ab)^n = a^n b^n, b0=1b^0 = 1.
  • b−n=1bnb^{-n} = \frac{1}{b^n}: a negative exponent is a reciprocal, never a negative number.
  • bm/n=(bn)mb^{m/n} = (\sqrt[n]{b})^m: the denominator is the root, the numerator the power. Take the root first to keep numbers small.

Part 6 · Misconception

A common mistake

The wrong idea: 2−3=−82^{-3} = -8.

What actually happens: 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}. A negative exponent flips the number into a fraction; it does not change the sign.

Part 7 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. Evaluate 82/38^{2/3} without a calculator.

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

The denominator 33 is a cube root and the numerator 22 is a power: 82/38^{2/3}=(83)2= (\sqrt[3]{8})^2=22= 2^2=4= 4.

  • Answer: 4

2. Simplify (x3)2⋅x−4(x^3)^2 \cdot x^{-4} to a single power of xx.

Type an expression.

Show the answer

Power of a power multiplies: (x3)2=x6(x^3)^2 = x^6. Same base multiplies by adding exponents: x6⋅x−4x^6 \cdot x^{-4}=x6+(−4)= x^{6 + (-4)}=x2= x^2.

  • Answer: x2x^{2}

3. Evaluate 16−1/216^{-1/2} exactly.

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

16−1/216^{-1/2}=1161/2= \frac{1}{16^{1/2}}=116= \frac{1}{\sqrt{16}}=14= \frac{1}{4}.

  • Answer: 14\frac{1}{4} ≈ 0.250

4. Solve 4x=324^x = 32 by writing both sides as powers of 22.

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

4x=(22)x=22x4^x = (2^2)^x = 2^{2x} and 32=2532 = 2^5. Equal powers of the same base have equal exponents: 2x=52x = 5, so x=52x = \frac{5}{2}.

  • Answer: 52\frac{5}{2} ≈ 2.500

5. A student writes 23⋅24=472^3 \cdot 2^4 = 4^7. What is the error?

  1. The exponents should be multiplied, giving 2122^{12}
  2. The base stays 22; only the exponents are added, giving 272^7
  3. The bases should be added, giving 4124^{12}
  4. There is no error; 47=23⋅244^7 = 2^3 \cdot 2^4
Show the answer

When you multiply powers of the same base, keep the base and add the exponents: 23⋅24=27=1282^3 \cdot 2^4 = 2^7 = 128. The student's 47=16,3844^7 = 16{,}384 is far too large.

  • The exponents should be multiplied, giving 2122^{12}: Multiplying exponents is for a power of a power, like (23)4(2^3)^4.
  • Correct: The base stays 22; only the exponents are added, giving 272^7: Keep the base, add the exponents.
  • The bases should be added, giving 4124^{12}: Bases are never combined by this rule, and exponents are added, not multiplied.
  • There is no error; 47=23⋅244^7 = 2^3 \cdot 2^4: 27=1282^7 = 128 but 47=16,3844^7 = 16{,}384.

Part 8 · Summary

Summary

Same base: add exponents to multiply, subtract to divide, multiply exponents for a power of a power. b−nb^{-n} is a reciprocal and bm/nb^{m/n} is a root then a power. Writing both sides with one base turns 4x=324^x = 32 into 2x=52x = 5.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections

Question set for this topic