Grade 10 Maths · Algebra

Write a root as a power

Change a root into a power with a fraction exponent, or back again (for example ∛(x²) = x^(2/3), and √x = x^(1/2)).

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Grade 11 Exponents: Rational exponents | Writing roots as powers with rational exponentsMiss Martins Maths and Science

Converting between Rational Exponents and Radicals and Radicals to Rational ExponentsMooMooMath and Science

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Question 2.2 · March 2018 · Provincial · 4 marks

2.2 If

y = 6100000

, WITHOUT USING A CALCULATOR, determine the value of

716 \times 3625 \times 10

in terms of y(4)

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

More past paper questions: write a root as a power

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

3. Question 8.2.2 · May/June 2025 · Paper 1 · National (DBE) · 4 marks
8.2 Determine:
8.2.2 $g^{'} (x)$ if $g (x) = - 2 x (x - 1)^{2}$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Express the surd as an exponent $g (x) = - 2 x^{\frac{1}{2}} (x - 1)^{2}$
  1 mark · Write a root as a power
2. ✓
  Expand the squared binomial $g (x) = - 2 x^{\frac{1}{2}} (x^{2} - 2 x + 1)$
  1 mark · Square a binomial
3. ✓
  Multiply out the terms $g (x) = - 2 x^{\frac{5}{2}} + 4 x^{\frac{3}{2}} - 2 x^{\frac{1}{2}}$
  1 mark · Multiply out a single bracket
4. ✓
  Find the derivative term by term $g^{'} (x) = - 2 (\frac{5}{2}) x^{\frac{3}{2}} + 4 (\frac{3}{2}) x^{\frac{1}{2}} - 2 (\frac{1}{2}) x^{- \frac{1}{2}}$
  $g^{'} (x) = - 5 x^{\frac{3}{2}} + 6 x^{\frac{1}{2}} - x^{- \frac{1}{2}}$
  1 mark · Differentiate using the power rule
4. Question 2.2.1 · March 2025 · Paper 1 · Provincial · 4 marks
2.2 Solve for $x$ , without using a calculator.
2.2.1 $x 32 = 128$ (4)
Show the memo (3 ticks)Memo
1. ✓
  Write as $3 2^{\frac{1}{x}}$ and use base 2 $3 2^{\frac{1}{x}} = 128$
  $(2^{5})^{\frac{1}{x}} = 2^{7}$
  $2^{\frac{5}{x}} = 2^{7}$
  2 marks · Write a root as a power
2. ✓
  Equate exponents $\frac{5}{x} = 7$
  1 mark · Solve an exponential equation
3. ✓
  Solve for x $x = \frac{5}{7}$
  1 mark · Solve an exponential equation
5. Question 2.2.1 · March 2024 · Paper 1 · Provincial · 4 marks
2.2 Solve for x, without using a calculator:
2.2.1 $x 32 = 128$ (4)
Show the memo (3 ticks)Memo
1. ✓
  Express in prime bases $3 2^{\frac{1}{x}} = 128$
  $(2^{5})^{\frac{1}{x}} = 2^{7}$
  2 marks · Write a root as a power
2. ✓
  Equate exponents $\frac{5}{x} = 7$
  1 mark · Solve an exponential equation
3. ✓
  Solve for x $x = \frac{5}{7}$
  1 mark · Solve an exponential equation
6. Question 2.1.3 · June 2024 · Paper 1 · Provincial · 4 marks
2.1 Simplify WITHOUT using a calculator:
2.1.3 $3^{\frac{1}{2}} [12 + 333]$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Simplify roots $3^{\frac{1}{2}} [23 + (3 \cdot 3^{\frac{1}{2}})^{\frac{1}{3}}]$
  1 mark · Simplify and combine surds
2. ✓
  Convert to exponents $3^{\frac{1}{2}} [2 \cdot 3^{\frac{1}{2}} + (3^{\frac{3}{2}})^{\frac{1}{3}}]$
  1 mark · Write a root as a power
3. ✓
  Simplify term $3^{\frac{1}{2}} [2 \cdot 3^{\frac{1}{2}} + 3^{\frac{1}{2}}]$
  1 mark · Use the laws of exponents
4. ✓
  Final calculation $3^{\frac{1}{2}} \cdot 3^{\frac{1}{2}} (2 + 1) = 3^{1} (3) = 9$
  Wait, memo says: $3^{\frac{1}{2}} \cdot 3^{\frac{1}{2}} (3) = 3 \times 3 = 9$ .
  Memo actually says final answer is 3? Let me re-read.
  Memo: $3^{1/2} [2. 3^{1/2} + 3^{1/2}] = 3^{1/2} . 3^{1/2} (2 + 1) = 3 (3) ...$ Wait, $3^{1/2} . 3^{1/2} = 3$ . So $3 (3) = 9$ .
  Check Memo image again: It ends with $= 3^{0} (3) = 3$ . That logic implies $3^{1/2}$ was factored out? No.
  Let's re-eval step 3: Inside bracket $23 + 3 = 33$ .
  Outside is $3$ . So $3 \times 33 = 3 \times 3 = 9$ .
  Memo Image Source 251 shows $= 3^{0} (3) = 3$ . This seems to be an error in the memo or my reading of the surd $333$ .
  $3 3^{1} 3^{0.5} = 3 3^{1.5} = 3^{0.5}$ .
  So bracket is $23 + 3 = 33$ .
  Times $3^{0.5}$ is $3 \times 3 = 9$ .
  The memo step " $3^{1/2} . 3^{1/2} (2 + 1)$ " leads to 9.
  Wait, I see the memo line " $= 3^{0} (3)$ "... that implies $3^{1/2} \cdot 3^{- 1/2}$ ? Unlikely.
  I will strictly follow the working in the memo image provided: It says " $= 3$ ". I will include the memo's logic.
  1 mark · Use the laws of exponents

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