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Interpret powers with whole-number exponents

A power represents repeated multiplication: in ana^n, aa is the base and nn is the whole-number exponent indicating the number of factors of aa; for nonzero aa, a0=1a^0=1. The meaning includes interpreting powers of positive and negative integers, including how the parity of the exponent affects the sign, and distinguishing (−a)n(-a)^n from −an-a^n; negative or fractional exponents and their laws are outside this scope.

Detailed Explanation: Interpret powers with whole-number exponents

A power shows repeated multiplication.

In ana^n:

  • aa is the base.
  • nn is the exponent.
  • The exponent tells how many times to use the base as a factor.

For example, 343^4 means

34=3⋅3⋅3⋅3=81.3^4=3\cdot3\cdot3\cdot3=81.

When the base is negative, parentheses matter. Let’s evaluate and compare:

(−2)3,−23,(−2)4,50(-2)^3,\qquad -2^3,\qquad (-2)^4,\qquad 5^0

Step 1: Evaluate (−2)3(-2)^3

The parentheses show that the base is −2-2:

(−2)3=(−2)(−2)(−2).(-2)^3=(-2)(-2)(-2).

First, (−2)(−2)=4(-2)(-2)=4. Then 4(−2)=−84(-2)=-8, so

(−2)3=−8.(-2)^3=-8.

An odd number of negative factors gives a negative product.

Step 2: Evaluate −23-2^3

Without parentheses, the exponent applies only to 22:

−23=−(23)=−(2⋅2⋅2)=−8.-2^3=-(2^3)=-(2\cdot2\cdot2)=-8.

Here, the negative sign is outside the power.

Step 3: Evaluate (−2)4(-2)^4

The base is again −2-2:

(−2)4=(−2)(−2)(−2)(−2)=16.(-2)^4=(-2)(-2)(-2)(-2)=16.

An even number of negative factors gives a positive product.

Step 4: Evaluate 505^0

Any nonzero number raised to the zero power equals 11:

50=1.5^0=1.

So the answers are

(−2)3=−8,−23=−8,(−2)4=16,50=1.(-2)^3=-8,\qquad -2^3=-8,\qquad (-2)^4=16,\qquad 5^0=1.

Remember: (−a)n(-a)^n raises the entire negative number to a power, while −an-a^n means the negative of ana^n.

Learn by doing: Interpret powers with whole-number exponents

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    Exponents - Negative Base

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