Grade 10 Maths · Algebra

Use the laws of exponents

Use the laws of powers, like adding the powers when you multiply the same base (2³ × 2² = 2⁵) and knowing that x⁰ = 1.

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Exponents Grade 8 | All Laws CombinedKevinmathscience

Grade 9 Exponents: Negative ExponentsKevinmathscience

Simplifying Exponents With Fractions, Variables, Negative Exponents, Multiplication & Division, MathThe Organic Chemistry Tutor

Algebra Basics: Laws Of Exponents - Math Anticsmathantics

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Question 5.1 · June 2024 · Paper 1 · Provincial · 2 marks

The graph of

f (x) = 2^{x + 2} + k

is drawn below. Diagram for this question

5.1 Use the laws of indices to show that

2^{x + 2} + k

is the same as

4. 2^{x} + k

.(2)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: use the laws of exponents

Work each one out first, then open the memo and check every tick.

3. Question 1.3.1 · March 2021 · Provincial · 3 marks
1.3 Simplify the following expressions fully:
1.3.1 $25 x^{4} y^{- 2} \times (5 x^{3} y^{0})^{- 2}$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Apply the negative exponent to the bracket $25 x^{4} y^{- 2} \times 5^{- 2} x^{- 6} y^{0}$
  1 mark · Use the laws of exponents
2. ✓
  Simplify the indices $5^{2 - 2} x^{4 - 6} y^{- 2 + 0} = 5^{0} x^{- 2} y^{- 2}$
  1 mark · Use the laws of exponents
3. ✓
  Answer $\frac{1}{x ^{2} y ^{2}}$
  1 mark · Use the laws of exponents
4. Question 1.1.4 · November 2024 · Paper 1 · National (DBE) · 5 marks
1.1 Solve for $x$ :
1.1.4 $2^{2 x} - 2^{x + 2} - 32 = 0$ (5)
Show the memo (4 ticks)Memo
1. ✓
  Expand exponential term $2^{2 x} - 2^{2} \cdot 2^{x} - 32 = 0$
  $2^{2 x} - 4 \cdot 2^{x} - 32 = 0$
  1 mark · Use the laws of exponents
2. ✓
  Factorise $(2^{x} - 8) (2^{x} + 4) = 0$
  1 mark · Factorise a trinomial
3. ✓
  Solve for the exponential term $2^{x} = 8$ or $2^{x} = - 4$
  2 marks · Solve an exponential equation
4. ✓
  Solve for x $2^{x} = 2^{3}$
  $x = 3$
  1 mark · Solve an exponential equation
5. Question 2.1.2 · March 2025 · Provincial · 4 marks
2.1 Evaluate the following expressions, without using a calculator:
2.1.2 $\frac{8 ^{k} \cdot 6 ^{k - 3} \cdot 9 ^{1 - k}}{1 6 ^{k - 1} \cdot 3 ^{- k}}$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Convert to prime bases $\frac{( 2 ^{3} ) ^{k} \cdot ( 2 \cdot 3 ) ^{k - 3} \cdot ( 3 ^{2} ) ^{1 - k}}{( 2 ^{4} ) ^{k - 1} \cdot 3 ^{- k}}$
  1 mark · Change the base of a power
2. ✓
  Applying the exponential laws correctly $= \frac{2 ^{3 k} \cdot 2 ^{k - 3} \cdot 3 ^{k - 3} \cdot 3 ^{2 - 2 k}}{2 ^{4 k - 4} \cdot 3 ^{- k}}$
  1 mark · Use the laws of exponents
3. ✓
  Simplification $= 2^{3 k + k - 3 - 4 k + 4} \cdot 3^{k - 3 + 2 - 2 k + k}$
  $= 2^{1} \cdot 3^{- 1}$
  1 mark · Use the laws of exponents
4. ✓
  Answer $= \frac{2}{3}$
  1 mark · Use the laws of exponents
6. Question 1.3.4 · March 2025 · Provincial · 4 marks
1.3 Simplify the following fully:
1.3.4 $\frac{1 0 ^{2 x + 3} \cdot 4 ^{1 - x}}{2 5 ^{2 + x}}$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Convert to prime bases $\frac{( 2 \cdot 5 ) ^{2 x + 3} \cdot ( 2 ^{2} ) ^{1 - x}}{( 5 ^{2} ) ^{2 + x}}$
  $= \frac{2 ^{2 x + 3} \cdot 5 ^{2 x + 3} \cdot 2 ^{2 - 2 x}}{5 ^{4 + 2 x}}$
  1 mark · Change the base of a power
2. ✓
  Group bases $2^{2 x + 3 + 2 - 2 x} \cdot 5^{2 x + 3 - (4 + 2 x)}$
  1 mark · Use the laws of exponents
3. ✓
  Simplify exponents $2^{5} \cdot 5^{- 1}$
  1 mark · Use the laws of exponents
4. ✓
  Final Answer $\frac{32}{5}$
  1 mark · Use the laws of exponents

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