Exponents and Roots

Powers and Bases

Evaluate whole-number powers, including the sign difference between -b² and (-b)².

powers-and-bases For a power b^n: - Multiply b by itself n times. - -b^n = -(b^n): the exponent applies before the negative sign. - (-b)^n: the negative base is raised to n.

Example

Read powers as repeated multiplication and watch how signs behave.

highlighted = computed this step

Step 1 — Set up a power

Set up the expression.

252^{5}
2^5 means 5 copies of the base2^522222The exponent counts the factor boxes.

Step 2 — Expand repeated factors

Write 2^5 as repeated multiplication.

25=2×2×2×2×2\hlmath{2^{5}} = 2 \times 2 \times 2 \times 2 \times 2
Write 2^5 as repeated multiplication2^52x2x2x2x25 factors, each equal to 2.

Step 3 — Multiply the factors

Multiply the 5 factors of 2: 2^5 = 32.

2×2×2×2×2=322 \times 2 \times 2 \times 2 \times 2 = \hl{32}
Multiply the 5 factors to get 322^52x2x2x2x22 x 2 x 2 x 2 x 2 = 322^5 = 32

Step 4 — No parentheses

Without parentheses, the negative stays outside: -(3^2) = -9.

−32=−(32)=-9- 3^{2} = -( 3^{2} )= \hl{-9}
Without parentheses: -(3^2) = -9negative outside-3^2= -9The minus is applied after the power.

Step 5 — Parentheses change the base

With parentheses, the base is -3: (-3)^2 = 9.

(−3)2=9( -3 )^{ 2 }= \hl{9}
With parentheses: (-3)^2 = 9negative outside-3^2= -9negative inside base-32= 9Parentheses make the negative part of each factor.

Step 6 — Result

Read the final result.

32−9932 \quad -9 \quad 9
Parentheses decide the basenegative outside-3^2= -9negative inside base-32= 9Parentheses make the negative part of each factor.