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

Question 14

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:

$a^m \times a^n = a^{m+n}$

Applying this rule:

$a^4 \times a^3 = a^{4 + 3} = a^7$

Thus, we have:

$a^x = a^7$

This implies that:

$x = 7$ .

Step 2

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

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Answer

Using the power of a power rule, which states:

$(a^m)^n = a^{m \cdot n}$

We can solve for $k$ as follows:

$(b^4)^3 = b^{4 \cdot 3} = b^{12}$

Therefore, from the equation:

$b^k = b^{12}$

It follows that:

$k = 12$ .

Step 3

(b) Factorise fully.

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Answer

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

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

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

This is the fully factored form.

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