Learn: Surds & Advanced Exponents

Review the key ideas, then practise whenever you’re ready.

The idea

A square root is the non-negative number that squares to give the number under the root: √25 = 5. A surd is an irrational root, such as √3. First review powers, then see how they connect to roots.

Key ideas

1
Powers with the same base - When multiplying powers with the same base, add the exponents. When dividing, subtract them: $a^m \times a^n=a^{m+n}$ and $a^m \div a^n=a^{m-n}$ , where $a\ne0$ . A power raised to another power multiplies the exponents: $(a^m)^n=a^{mn}$ .
2
Powers over multiplication and division - A power applies to each factor: $(ab)^n=a^n b^n$ and $(a/b)^n=a^n/b^n$ , where $b\ne0$ . It does not distribute over addition: $(2+3)^2=25$ , but $2^2+3^2=13$ .
3
Zero and negative exponents - For $a\ne0$ , $a^0=1$ and $a^{-n}=1/a^n$ . A negative exponent means take the reciprocal; it does not make the value negative.
4
Roots and surds - To simplify a square root, factor out a perfect square: $\sqrt{75}=\sqrt{25 \times 3}=5\sqrt{3}$ , because $\sqrt{25}=5$ .
5
Multiplying and dividing roots - For $a,b\ge0$ , $\sqrt a\,\sqrt b=\sqrt{ab}$ . Also, $\frac{\sqrt a}{\sqrt b}=\sqrt{\frac ab}$ when $b>0$ . Roots do not generally distribute over addition: $\sqrt{9+16}\ne\sqrt9+\sqrt{16}$ , since $5\ne7$ .
6
Fractional exponents - For $a>0$ and positive integers $m,n$ , $a^{m/n}=\sqrt[n]{a^m}$ . The denominator gives the root; the numerator gives the power. For example, $8^{2/3}=(\sqrt[3]{8})^2=4$ .
7
Combining surds - Simplify each root first. Then combine terms with the same root: $\sqrt{50}+\sqrt{18}=5\sqrt2+3\sqrt2=8\sqrt2$ .
8
Rationalise denominators - Multiply the numerator and denominator by the same non-zero expression, so the fraction stays equal. For a two-term denominator, use its conjugate: the same terms with the sign changed, such as $3-\sqrt7$ and $3+\sqrt7$ .
9
Equations with powers and roots - Use exponent rules to solve equations with matching bases. If you square both sides, check your answer in the original equation.

Interactive Surds & Fractional Power Explorer

Pick a tab to see how roots and fractional powers work.

Write $\sqrt{75} = \sqrt{25 \times 3}$ . Only 25 = 5² is a perfect square, so $\sqrt{25} = 5$ steps outside the root. The 3 has no square partner, so it stays inside.

\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25}\times\sqrt{3} = 5\sqrt{3}

Perfect squares to hunt for: 4, 9, 16, 25, 36, 49, 64, 81, 100

Worked example

Every step explained - why we do it, and what happens.

1
$\sqrt{50} + \sqrt{18}$
Why: We cannot combine these roots yet because their root parts differ. Simplify each one first by finding a perfect-square factor.
What happens: Look for a perfect square that divides both 50 and 18.
2
$= \sqrt{25 \times 2} + \sqrt{9 \times 2}$
Why: Write each number as a perfect square times 2: 50 = 25 × 2 and 18 = 9 × 2.
What happens: Now each root has a perfect-square factor we can simplify.
3
$= \sqrt{25} \cdot \sqrt{2} + \sqrt{9} \cdot \sqrt{2}$
Why: For non-negative factors, the square root of a product equals the product of the square roots.
What happens: This lets us take the square roots of 25, 9, and 2 separately.
4
$= 5\sqrt{2} + 3\sqrt{2}$
Both terms now have the same root part, √2.
5
$= (5 + 3)\sqrt{2} = 8\sqrt{2}$
Why: Add the coefficients (5 + 3 = 8) and keep √2 unchanged, just as 5x + 3x = 8x.
What happens: The simplified result is 8√2.

Rationalizing Denominators & Matching Bases (Optional Toolbox)

How to clear square roots from the bottom of a fraction and solve exponential equations like $4^{x+1} = 8^2$ .

Optional - Extension Level

1. Conjugate Trick for Denominators

To clear $3 - \sqrt{7}$ from the denominator in the fraction shown below, multiply the numerator and denominator by its conjugate $3 + \sqrt{7}$ . The denominator becomes $3^2 - (\sqrt{7})^2 = 9 - 7 = 2$ because $(a-b)(a+b)=a^2-b^2$ (the root is gone):

\frac{6}{3 - \sqrt{7}} \times \frac{3 + \sqrt{7}}{3 + \sqrt{7}} = \frac{6(3 + \sqrt{7})}{2} = 3(3 + \sqrt{7}) = 9 + 3\sqrt{7}

2. Matching Bases for Power Equations

In $4^{x+1} = 8^2$ , spot that 4 = 2² and 8 = 2³. Rewrite both sides with base 2, then equate the exponents:

(2^2)^{x+1} = (2^3)^2 \;\Rightarrow\; 2(x+1) = 6 \;\Rightarrow\; x = 2

Real-world example

Two square skylights sit side by side along a straight roof beam. Their areas are 75 m² and 12 m². What is their combined width in simplest surd form?

1
$W = \sqrt{75}\,\text{m} + \sqrt{12}\,\text{m}$
Why: The side length of a square is the square root of its area.
What happens: We need to simplify √75 and √12 and add them.
2
$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \text{ m}$
Why: Since 75 = 25 × 3 and √25 = 5, the side length is 5√3 m.
What happens: The large skylight is 5√3 m wide.
3
$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \text{ m}$
Why: Since 12 = 4 × 3 and √4 = 2, the side length is 2√3 m.
What happens: The small skylight is 2√3 m wide.
4
$\text{Trap: } \sqrt{75} + \sqrt{12} = \sqrt{87}\,?$
Why: You cannot add the numbers inside separate square roots. √87 is about 9.33 m, but the correct total is about 12.12 m.
What happens: Simplify each surd first, then add the matching root parts.
5
$5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3} \text{ m} \quad (k = 7)$
Why: Both terms have √3, so add their coefficients: 5 + 2 = 7.
What happens: The total width is 7√3 m.

Common glitch

A few mistakes to watch for:

1.Adding numbers directly under separate square roots

Wrong: $\sqrt{9} + \sqrt{16} = \sqrt{25} = 5$ - check it: $3 + 4 = 7$ , not $5$ !

Right:Simplify each root first, and only add the outside numbers when the root part matches.

2.Mixing up the power and the root in a fractional exponent

Wrong:Reading $8^{2/3}$ as “square root of $8^3$ ”

Right:The bottom number tells the root: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$ .

3.Thinking a negative exponent makes the answer negative

Wrong: $2^{-3} = -8$ (thinking the negative sign makes the answer negative)

Right:A negative exponent means reciprocal, not a negative value: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ .

Prime Factorization Calculator - Factor a number into primes to spot every perfect-square pair under a root.
General Calculator - Check roots and powers with the scientific calculator keys.

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Common glitch

A few mistakes to watch for:

1.Adding numbers directly under separate square roots

Wrong: $\sqrt{9} + \sqrt{16} = \sqrt{25} = 5$ - check it: $3 + 4 = 7$ , not $5$ !

Right:Simplify each root first, and only add the outside numbers when the root part matches.

2.Mixing up the power and the root in a fractional exponent

Wrong:Reading $8^{2/3}$ as “square root of $8^3$ ”

Right:The bottom number tells the root: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$ .

3.Thinking a negative exponent makes the answer negative

Wrong: $2^{-3} = -8$ (thinking the negative sign makes the answer negative)

Right:A negative exponent means reciprocal, not a negative value: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ .

Learn: Surds & Advanced Exponents

The idea

Key ideas

Worked example

Real-world example

Common glitch

Try a related calculator

Learn: Surds & Advanced Exponents

The idea

Key ideas

Worked example

Real-world example

Common glitch

Try a related calculator