You can use the laws of indices to simplify powers of the same base.

"Index" (plural: "indices") and "exponent" mean the same thing as "power".

The laws of indices:

• $a^x×a^y=a^{x+y}$
• $a^x÷a^y=a^{x-y}$
• $(a^x)^y=a^{xy}$
• $(ab)^x=a^xb^x$

Also remember:

• $a^0=1$

Example 1

Simplify $x^2 \times x^3$

$x^{2+3} = x^5$

Example 2

Simplify $x^7 \div x^4$

$x^{7-4} = x^3$

Example 3

Simplify $(x^3)^4$

$x^{3 \times 4} = x^{12}$

Example 4

Simplify $(2x)^4$

$2^4 \times x^4 = 16x^4$

Example 5

Simplify $\dfrac{x^5 \times x^2}{x^3}$

$\dfrac{x^{5+2}}{x^3} = \dfrac{x^7}{x^3} = x^{7-3} = x^4$

Example 6

Simplify $(x^2)^3 \times (x^4)^2$

$x^{2 \times 3} \times x^{4 \times 2} = x^6 \times x^8 = x^{6+8} = x^{14}$