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Section A (52 Marks) Answer all questions in this section 1 (a) Simplify 13a3/28a2b * 65a5b4 / 56a2b2 (b) Learners in a Grade 10 class were asked the types of drinks they liked. The Venn diagram shows their responses.

Mathematics

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Step-by-step solution

4 steps

  1. Combine the numerators and denominators of the given fractions.

    13a328b3×65ab56a4=13×65×a3×ab28×56×b3×a4\frac{13a^3}{28b^3} \times \frac{65a^b}{56a^4} = \frac{13 \times 65 \times a^3 \times a^b}{28 \times 56 \times b^3 \times a^4}

  2. Multiply the numerical coefficients.

    13×65=84513 \times 65 = 845 28×56=156828 \times 56 = 1568

    The expression becomes:

    845×a3×ab1568×b3×a4\frac{845 \times a^3 \times a^b}{1568 \times b^3 \times a^4}

  3. Simplify the terms with variable 'a' using the exponent rules xm×xn=xm+nx^m \times x^n = x^{m+n} and xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}.

    a3×aba4=a3+ba4=a(3+b)−4=ab−1\frac{a^3 \times a^b}{a^4} = \frac{a^{3+b}}{a^4} = a^{(3+b)-4} = a^{b-1}

  4. Combine all simplified parts.

    The term b3b^3 remains in the denominator.

    8451568×ab−1×1b3=845ab−11568b3\frac{845}{1568} \times a^{b-1} \times \frac{1}{b^3} = \frac{845 a^{b-1}}{1568 b^3}

    The simplified expression is:

    845ab−11568b3\boxed{\frac{845 a^{b-1}}{1568 b^3}}
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