Exponential Equations

Worked example

1. 1

   $2^x=32$ — The question. x is in the power, so dividing will not find it.

2. 2

   $2^x=2^5$ — Write 32 as a power of 2: 2 × 2 × 2 × 2 × 2 = 32, five 2s.

3. 3

   $x=5$ — Both sides are now 2 to a power, so the powers must be equal.

Worked example

1. 1

   $\frac{(5^{-2}{)}^3}{5^4}=5^n$ — The question. Every number is a power of 5, so aim for one power of 5 on the left.

2. 2

   $(5^{-2}{)}^3=5^{-6}$ — A bracket to a power: multiply the powers, (−2) × 3 = −6.

3. 3

   $\frac{5^{-6}}{5^4}=5^{-6-4}$ — Dividing powers of the same base: take the bottom power away from the top one.

4. 4

   $5^{-10}=5^n$ — −6 − 4 = −10: start at −6 and go 4 further down. Now there is one power on each side.

5. 5

   $n=-10$ — Same base on both sides, so the powers are equal.

Worked example

1. 1

   $9^x=27^2$ — The question. 9 and 27 are different bases, so you cannot match the powers yet.

2. 2

   $(3^2{)}^x=(3^3{)}^2$ — 9 is 3² and 27 is 3³, so both sides become powers of 3.

3. 3

   $3^{2x}=3^6$ — A bracket to a power: multiply the powers, 2 × x = 2x and 3 × 2 = 6.

4. 4

   $2x=6$ — The bases now match, so the powers must be equal.

5. 5

   $x=3$ — Divide both sides by 2.

Worked example

1. 1

   $9^x=\frac{1}{27}$ — The question. A fraction on the right, and two different bases.

2. 2

   $(3^2{)}^x=\frac{1}{3^3}$ — 9 is 3² and 27 is 3³.

3. 3

   $3^{2x}=\frac{1}{3^3}$ — A bracket to a power: multiply, 2 × x = 2x.

4. 4

   $3^{2x}=3^{-3}$ — One over $3^3$ is $3^{-3}$: that is what a negative power means.

5. 5

   $2x=-3$ — Same base on both sides, so the powers are equal.

6. 6

   $x=-\frac{3}{2}$ — Divide by 2. A negative answer is fine: x is a power, not a length.

Worked example

1. 1

   $\frac{3^x}{9^{x+1}}=27$ — The question. The bases 3, 9 and 27 are all powers of 3, so change everything to base 3.

2. 2

   $9^{x+1}=(3^2{)}^{x+1}$ — 9 is 3².

3. 3

   $(3^2{)}^{x+1}=3^{2x+2}$ — A bracket to a power: multiply. The 2 multiplies the whole of x + 1, giving 2x + 2.

4. 4

   $\frac{3^x}{3^{2x+2}}=3^3$ — Put that back in, and write 27 as $3^3$.

5. 5

   $3^{x-(2x+2)}=3^3$ — Dividing: subtract the powers. Keep the bracket, because the whole of 2x + 2 is taken away.

6. 6

   $x-(2x+2)=3$ — Same base on both sides, so the powers are equal.

7. 7

   $x-2x-2=3$ — Open the bracket. The minus reaches both terms: −(2x + 2) = −2x − 2.

8. 8

   $-x-2=3$ — Collect the x terms: x − 2x = −x.

9. 9

   $-x=5$ — Add 2 to both sides.

10. 10

    $x=-5$ — Multiply both sides by −1 to change the sign.

Worked example

1. 1

   $4^{x+1}={\left(\frac{1}{8}\right)}^{x-3}$ — The question. x is in both powers, and the bases 4 and ⅛ are different.

2. 2

   $(2^2{)}^{x+1}=(2^{-3}{)}^{x-3}$ — 4 is 2². ⅛ is 1 over 2³, which is $2^{-3}$.

3. 3

   $2^{2(x+1)}=2^{-3(x-3)}$ — A bracket to a power: multiply. Keep the brackets so the −3 reaches both terms.

4. 4

   $2^{2x+2}=2^{-3x+9}$ — Expand both. On the right, (−3) × x = −3x and (−3) × (−3) = +9.

5. 5

   $2x+2=-3x+9$ — Same base on both sides, so the powers are equal.

6. 6

   $5x+2=9$ — Add 3x to both sides, so all the x terms are on the left.

7. 7

   $5x=7$ — Take 2 from both sides.

8. 8

   $x=\frac{7}{5}=1.4$ — Divide by 5. A fraction or decimal answer is fine.

Worked example

1. 1

   $9^m\times 3=27^{2n}$ — The question: find m in terms of n. The answer will have n in it.

2. 2

   $(3^2{)}^m\times 3=(3^3{)}^{2n}$ — 9 is 3² and 27 is 3³.

3. 3

   $3^{2m}\times 3^1=3^{6n}$ — A bracket to a power: multiply, 2 × m and 3 × 2n. A plain 3 is $3^1$.

4. 4

   $3^{2m+1}=3^{6n}$ — Multiplying powers of the same base: add the powers.

5. 5

   $2m+1=6n$ — Same base on both sides, so the powers are equal.

6. 6

   $2m=6n-1$ — Take 1 from both sides to get the m term alone.

7. 7

   $m=\frac{6n-1}{2}$ — Divide both sides by 2. This is the answer: n stays in it.

Worked example

1. 1

   $\frac{10^n\times 2^{n^2-4n}}{5^n}=16$ — The question. 10 and 5 are not powers of 2, so split 10 into primes first.

2. 2

   $10^n=2^n\times 5^n$ — 10 is 2 × 5, and the power n goes on both parts.

3. 3

   $2^n\times 2^{n^2-4n}=16$ — The 5ⁿ on top and bottom cancel, leaving only powers of 2.

4. 4

   $2^n\times 2^{n^2-4n}=2^4$ — 16 = 2 × 2 × 2 × 2, so 16 is $2^4$.

5. 5

   $2^{n+n^2-4n}=2^4$ — Multiplying powers of the same base: add the powers.

6. 6

   $n+n^2-4n=4$ — Same base on both sides, so the powers are equal.

7. 7

   $n^2-3n=4$ — Collect the n terms: n − 4n = −3n.

8. 8

   $n^2-3n-4=0$ — Take 4 from both sides, so the quadratic equals 0 and can be factorised.

9. 9

   $(n-4)(n+1)=0$ — Find two numbers that multiply to −4 and add to −3: they are −4 and +1.

10. 10

    $n=4\text{or}n=-1$ — One bracket must be 0: n − 4 = 0 or n + 1 = 0. Give both answers.

Worked example

1. 1

   $x^{\frac{1}{2}}=8x^{-1}$ — The question. x is the number here, and it appears twice.

2. 2

   $x^{\frac{1}{2}}\times x=8$ — Multiply both sides by x, because $x^{-1}\times x=1$.

3. 3

   $x^{\frac{3}{2}}=8$ — Multiplying: add the powers, ½ + 1 = 3/2. Now x appears once.

4. 4

   $x=8^{\frac{2}{3}}$ — Use the flipped power 2/3 on both sides.

5. 5

   $\sqrt[3]{8}=2$ — The bottom 3 means cube root. Do the root first so the number stays small.

6. 6

   $x=2^2=4$ — The top 2 means square. Check: √4 = 2 and 8 ÷ 4 = 2.