10.2.2 Exponential equations · Year 10 Extended · Unit 2: Exponents and Surds

Write it with the same base

Board · L04 · Do now

Do now

Write (a) to (d) as a power of 2 and (e) to (h) as a power of 3.

(a) \(8\)
(b) \(\frac{1}{4}\)
(c) \(\sqrt{2}\)
(d) \(0.125\)
(e) \(\frac{1}{27}\)
(f) \(\sqrt[3]{3}\)
(g) \(9^x\)
(h) \(\left(\frac{1}{9}\right)^x\)

(a) \(8 = 2^3\)
(b) \(\frac{1}{4} = 2^{-2}\)
(c) \(\sqrt{2} = 2^{1/2}\)
(d) \(0.125 = 2^{-3}\)
(e) \(\frac{1}{27} = 3^{-3}\)
(f) \(\sqrt[3]{3} = 3^{1/3}\)
(g) \(9^x = 3^{2x}\)
(h) \(\left(\frac{1}{9}\right)^x = 3^{-2x}\)

Board · L03 · Write it with the same base

Write it with the same base

As a power of 2

\(8 =\)
\(\frac{1}{4} =\)
\(\sqrt{2} =\)
\(4\sqrt{2} =\)
\(\frac{1}{\sqrt{8}} =\)

As a power of 3

\(27 =\)
\(\frac{1}{9} =\)
\(\sqrt{3} =\)
\(9\sqrt{3} =\)

If \(a^p = a^q\), then \(p = q\). Same base \(\Rightarrow\) equal indices.

Same base

Board · L03 · Worked example 1

Bases already match

(a)

Solve \(2^x = 32\)

\(2^x = 2^5\)
\(x = 5\)

(b)

Solve \(3^{x+1} = 81\)

\(3^{x+1} = 3^4\)
\(x + 1 = 4 \;\Rightarrow\; x = 3\)

Board · L03 · Worked example 2

Common base

Example

Solve \(27^n = 9\). Write both sides as powers of 3.

\((3^3)^n = 3^2\)
\(3^{3n} = 3^2\)
\(3n = 2 \;\Rightarrow\; n = \frac{2}{3}\)

Board · L03 · Worked example 3

Common base

Example

\(64^m = 32\). We also know that \(2^5 = 32\). Find the value of \(m\).

\((2^6)^m = 2^5\)
\(2^{6m} = 2^5\)
\(6m = 5 \;\Rightarrow\; m = \frac{5}{6}\)

Board · L03 · Worked example 4

Products

Example

Solve \(4^x \times 8^{2-x} = 2^x \times 4^{x-1}\).

\(2^{2x} \times 2^{6-3x} = 2^x \times 2^{2x-2}\)
\(2^{6-x} = 2^{3x-2}\)
\(6 - x = 3x - 2\)
\(8 = 4x \;\Rightarrow\; x = 2\)

Examples 1 to 13

Board · L04 · Example 1

Same base, same index

Key idea

If \(a^m = a^n\), then \(m = n\).

Example

Solve \(2^x = 8\)

\(2^x = 2^3\)
\(x = 3\)

Check \(2^3 = 8\)

Your turn

Solve \(3^x = 81\)

\(3^x = 3^4\)
\(x = 4\)

Check \(3^4 = 81\)

Board · L04 · Example 2

Unknown on both sides

Example

Solve \(5^{2x-1} = 5^{x+3}\)

\(2x - 1 = x + 3\)
\(x = 4\)

Check \(5^7 = 5^7\)

Your turn

Solve \(2^{4x-3} = 2^{x+6}\)

\(4x - 3 = x + 6\)
\(3x = 9\)
\(x = 3\)

Check \(2^9 = 2^9\)

Board · L04 · Example 3

Write the number as a power

Example

Solve \(3^{x-1} = 81\)

\(3^{x-1} = 3^4\)
\(x - 1 = 4\)
\(x = 5\)

Check \(3^4 = 81\)

Your turn

Solve \(2^{2x+1} = 32\)

\(2^{2x+1} = 2^5\)
\(2x + 1 = 5\)
\(x = 2\)

Check \(2^5 = 32\)

Board · L04 · Example 4

Negative indices

Example

Solve \(2^x = \frac{1}{16}\)

\(2^x = 2^{-4}\)
\(x = -4\)

Check \(2^{-4} = \frac{1}{16}\)

Your turn

Solve \(3^{2x} = \frac{1}{27}\)

\(3^{2x} = 3^{-3}\)
\(2x = -3\)
\(x = -\frac{3}{2}\)

Check \(3^{-3} = \frac{1}{27}\)

Board · L04 · Example 5

Change to a common base

Example

Solve \(8^x = 32\)

\(2^{3x} = 2^5\)
\(3x = 5\)
\(x = \frac{5}{3}\)

Check \(8^{5/3} = 2^5 = 32\)

Your turn

Solve \(4^{x+1} = 8\)

\(2^{2(x+1)} = 2^3\)
\(2x + 2 = 3\)
\(x = \frac{1}{2}\)

Check \(4^{3/2} = 8\)

Board · L04 · Example 6

Fraction bases

Example

Solve \(\left(\frac{1}{4}\right)^x = 32\)

\(2^{-2x} = 2^5\)
\(-2x = 5\)
\(x = -\frac{5}{2}\)

Check \(\left(\frac{1}{4}\right)^{-5/2} = 4^{5/2} = 32\)

Your turn

Solve \(\left(\frac{1}{27}\right)^x = 9^{x+1}\)

\(3^{-3x} = 3^{2(x+1)}\)
\(-3x = 2x + 2\)
\(x = -\frac{2}{5}\)

Check both \(3^{6/5}\)

Board · L04 · Example 7

Roots: fractional indices

Example

Solve \(\sqrt{8^x} = \frac{1}{4}\)

\(2^{3x/2} = 2^{-2}\)
\(\frac{3x}{2} = -2\)
\(x = -\frac{4}{3}\)

Check \(\sqrt{8^{-4/3}} = 2^{-2}\)

Your turn

Solve \(\sqrt[3]{25^x} = \frac{1}{\sqrt{5}}\)

\(5^{2x/3} = 5^{-1/2}\)
\(\frac{2x}{3} = -\frac{1}{2}\)
\(x = -\frac{3}{4}\)

Check both \(5^{-1/2}\)

Board · L04 · Example 8

Combine powers: multiply

Example

Solve \(2^x \times 2^3 = 2^7\)

\(2^{x+3} = 2^7\)
\(x + 3 = 7\)
\(x = 4\)

Check \(2^4 \times 2^3 = 16 \times 8 = 128\)

Your turn

Solve \(3^x \times 9 = 243\)

\(3^x \times 3^2 = 3^5\)
\(3^{x+2} = 3^5\)
\(x + 2 = 5\)
\(x = 3\)

Check \(27 \times 9 = 243\)

Board · L04 · Example 9

Combine powers: divide

Example

Solve \(5^x \div 5^2 = 5^4\)

\(5^{x-2} = 5^4\)
\(x - 2 = 4\)
\(x = 6\)

Check \(5^6 \div 5^2 = 5^4\)

Your turn

Solve \(2^x \div 4 = 32\)

\(2^x \div 2^2 = 2^5\)
\(2^{x-2} = 2^5\)
\(x - 2 = 5\)
\(x = 7\)

Check \(128 \div 4 = 32\)

Board · L04 · Example 10

Products and quotients

Example

Solve \(4^x \times 8^{x-1} = \frac{1}{2}\)

\(2^{2x} \times 2^{3x-3} = 2^{-1}\)
\(5x - 3 = -1\)
\(x = \frac{2}{5}\)

Check \(2^{4/5} \times 2^{-9/5} = 2^{-1}\)

Your turn

Solve \(\frac{27^x}{9^{x+1}} = \frac{1}{3}\)

\(3^{3x-2(x+1)} = 3^{-1}\)
\(x - 2 = -1\)
\(x = 1\)

Check \(\frac{27}{81} = \frac{1}{3}\)

Board · L04 · Example 11

Unknown base

Example

Solve \(y^{-3/2} = \frac{1}{8}\)

\(y = \left(\frac{1}{8}\right)^{-2/3}\)
\(= 8^{2/3} = 4\)

Check \(4^{-3/2} = \frac{1}{8}\)

Your turn

Solve \(y^{-3/4} = \frac{1}{27}\)

\(y = \left(\frac{1}{27}\right)^{-4/3}\)
\(= 27^{4/3} = 81\)

Check \(81^{-3/4} = \frac{1}{27}\)

Board · L04 · Example 12

Common factor

Example

Solve \(2^{x+2} + 2^x = 40\)

\(4 \times 2^x + 2^x = 40\)
\(5 \times 2^x = 40\)
\(2^x = 8 = 2^3\)
\(x = 3\)

Check \(32 + 8 = 40\)

Your turn

Solve \(3^{x+2} - 3^x = 72\)

\(9 \times 3^x - 3^x = 72\)
\(8 \times 3^x = 72\)
\(3^x = 9 = 3^2\)
\(x = 2\)

Check \(81 - 9 = 72\)

Board · L04 · Example 13

Squared index

Example

Solve \(27^{x^2} = 9^{x^2+2}\)

\(3^{3x^2} = 3^{2(x^2+2)}\)
\(3x^2 = 2x^2 + 4\)
\(x^2 = 4\)
\(x = \pm 2\)

Check both give \(3^{12} = 3^{12}\)

Your turn

Solve \(4^{x^2+1} = 8^{x^2-1}\)

\(2^{2(x^2+1)} = 2^{3(x^2-1)}\)
\(2x^2 + 2 = 3x^2 - 3\)
\(x^2 = 5\)
\(x = \pm\sqrt{5}\)

Check both give \(2^{12} = 2^{12}\)

Harder

Board · L03 · Harder

Harder

(a)

Given that \(n^{-4/5} = \left(\frac{1}{2}\right)^4\) and \(n > 0\), find \(n\).

\(n^{-4/5} = \frac{1}{16} \Rightarrow n^{4/5} = 16\)
\(n = 16^{5/4} = \left(\sqrt[4]{16}\right)^5 = 2^5 = 32\)

(b)

Given that \(\sqrt{k^{3a}} = k^{a+4}\), find \(a\).

\(k^{3a/2} = k^{a+4}\)
\(\frac{3a}{2} = a + 4 \Rightarrow 3a = 2a + 8 \Rightarrow a = 8\)

Board · L03 · Building to

Building to

(1)

Solve \(2^x \times 4^{x+1} = 8^3\)

(2)

Given that \(3^x + 3^{x+2} = 270\), find \(x\).

Board · L03 · Your turn

Your turn

Mini-whiteboards. Answer, then hold up.

Round 1: Solve \(81^p = 27\)
Round 2: Solve \(4^x = \frac{1}{8}\)
Round 3: Solve \(5^{2x-1} = 125\)

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By the end of this section

Worked examples

Write it with the same base

Do now

Do now

Write it with the same base

As a power of 2

As a power of 3

Same base

Bases already match

(a)

(b)

Common base

Example

Common base

Example

Products

Example

Examples 1 to 13

Same base, same index

Example

Your turn

Unknown on both sides

Example

Your turn

Write the number as a power

Example

Your turn

Negative indices

Example

Your turn

Change to a common base

Example

Your turn

Fraction bases

Example

Your turn

Roots: fractional indices

Example

Your turn

Combine powers: multiply

Example

Your turn

Combine powers: divide

Example

Your turn

Products and quotients

Example

Your turn

Unknown base

Example

Your turn

Common factor

Example

Your turn

Squared index

Example

Your turn

Harder

Harder

(a)

(b)

Building to

(1)

(2)

Your turn

Your turn