3.3 Laws of indices — Learn

Prerequisites: powers as repeated multiplication; multiplication and division of algebraic terms.

Objectives: simplify products and quotients of powers; simplify a power raised to a power; use zero and negative integer indices.

By the end of this section, you can use index laws to simplify products, quotients and powers, including zero and negative integer indices.

This section covers A4 and N7. An index is the small number that tells how many equal factors a power contains.

Multiply and divide powers

When the base stays the same, multiplying powers joins their factors. Dividing powers removes matching factors.

For non-zero $x$, $x^a x^b=x^{a+b}$ for non-negative integer indices. Also, $x^a\div x^b=x^{a-b}$ when $b>0$, with a non-zero divisor.

Example 1. Simplify $a^4a^3$.

A power raised to a power

$(x^a)^b$ means repeat the whole group $x^a$ a total of $b$ times. Therefore the indices multiply.

Example 2. Simplify $(y^3)^4$.

Zero and negative indices

For non-zero $x$, $x^0=1$. The quotient law explains this: $x^3\div x^3$ equals both 1 and $x^{3-3}=x^0$.

The same law gives $x^{-2}=1/x^2$, because $x^3\div x^5=x^{-2}$ and $x^3/x^5=1/x^2$.

Watch out. In $x^3x^4$, add the indices to get $x^7$. In $(x^3)^4$, multiply them to get $x^{12}$. The brackets show that the whole power repeats. Check by writing out the factors.

Progress check

Rewrite $2^{20}$ as $(2^{10})^2$. Use $2^{10}=1024$ to find the exact size after 20 doublings. Compare the result with your estimate from 3.2.

Try it: the laws of indices

Predict first. Without multiplying, predict $2^3\times2^2$ and $2^{3+2}$. Then calculate each expression to test your prediction.

Explain: why is $a^m\times a^n=a^{m+n}$? Describe how many factors of $a$ the product contains.

You should now be able to

• multiply and divide powers with the same base;
• simplify a power raised to a power;
• explain why a non-zero base to power zero equals 1;
• rewrite a negative integer index as a reciprocal;
• simplify algebraic products using index laws.