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14 (a) Find the value of x in each of the following - OCR - GCSE Maths - Question 14 - 2018 - Paper 1

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14 (a) Find the value of x in each of the following. (i) $a^4 \times a^3 = a^x$ (ii) $(b^4)^3 = b^k$ (b) Factorise fully. $18x^2 + 9x$

Worked Solution & Example Answer:14 (a) Find the value of x in each of the following - OCR - GCSE Maths - Question 14 - 2018 - Paper 1

Step 1

(i) $a^4 \times a^3 = a^x$

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Answer

To solve the equation, we apply the properties of exponents. According to the exponent product rule:

am×an=am+na^m \times a^n = a^{m+n}

Applying this rule:

a4×a3=a4+3=a7a^4 \times a^3 = a^{4 + 3} = a^7

Thus, we have:

ax=a7a^x = a^7

This implies that:

x=7x = 7.

Step 2

(ii) $(b^4)^3 = b^k$

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Answer

Using the power of a power rule, which states:

(am)n=amâ‹…n(a^m)^n = a^{m \cdot n}

We can solve for kk as follows:

(b4)3=b4â‹…3=b12(b^4)^3 = b^{4 \cdot 3} = b^{12}

Therefore, from the equation:

bk=b12b^k = b^{12}

It follows that:

k=12k = 12.

Step 3

(b) Factorise fully.

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Answer

To factorise the expression 18x2+9x18x^2 + 9x, we look for the greatest common factor (GCF) of the two terms:

The GCF of 18x218x^2 and 9x9x is 9x9x. Thus, we factor out 9x9x:

18x2+9x=9x(2x+1)18x^2 + 9x = 9x(2x + 1)

This is the fully factored form.

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