Grade 10

Algebraic Expressions

Master the language of algebra — exponents, surds, factorising and simplifying fractions.

Exponent Laws

An exponent (also called a power or index) tells you how many times to multiply a number by itself. For example, $2^3 = 2 \times 2 \times 2 = 8$ .

Theorem

\begin{aligned} a^m \cdot a^n &= a^{m+n} \\[4pt] \frac{a^m}{a^n} &= a^{m-n} \\[4pt] (a^m)^n &= a^{mn} \\[4pt] (ab)^n &= a^n b^n \\[4pt] \left(\frac{a}{b}\right)^n &= \frac{a^n}{b^n} \\[4pt] a^0 &= 1 \quad (a \ne 0) \\[4pt] a^{-n} &= \frac{1}{a^n} \end{aligned}

A negative exponent means "flip it" — put it in the denominator. It does NOT make the answer negative.

Worked Example

Simplify $\dfrac{2^5 \cdot 2^3}{2^4}$

Worked Example

Simplify $(3x^2)^3 \div 9x^4$

Worked Example

Write $5^{-2}$ and $\left(\frac{2}{3}\right)^{-1}$ without negative exponents.

Worked Example

Solve for $x$ : $3^{x+1} = 27$

Worked Example

Solve for $x$ : $4^x = 8$

Tip

When bases are the same and you multiply, ADD the exponents. When you divide, SUBTRACT the exponents. To solve exponential equations: make both sides the same base, then equate exponents.