Exercise solution 3, page 482 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) To decide whether a statement is true or false, you must simplify each side carefully and then compare the results. A statement is true only if both sides are equal for all allowed values of the variable.

B) When simplifying algebraic fractions, divide the numerical coefficients and simplify the powers of the same letter separately. For example,

\[ \frac{20a^4}{5a^4} = \frac{20}{5}\times \frac{a^4}{a^4} = 4\times 1 = 4 \]

This works because any non-zero quantity divided by itself is equal to \(1\).

C) For powers with the same base, use the index law

\[ \frac{x^m}{x^n} = x^{m-n} \]

So if the powers are the same, the result is

\[ \frac{x^m}{x^m} = x^{m-m} = x^0 = 1 \]

provided that \(x eq 0\).

D) Another important index law is the power of a power:

\[ (a^m)^n = a^{mn} \]

For example,

\[ (b^3)^2 = b^{3\times 2} = b^6 \]

This is useful when a bracket containing a power is itself raised to another power.

E) When a whole fraction is raised to a power, the power applies to both the numerator and the denominator:

\[ \left(\frac{p}{q}\right)^n = \frac{p^n}{q^n} \]

For example,

\[ \left(\frac{3c^2}{5}\right)^2 = \frac{(3c^2)^2}{5^2} = \frac{9c^4}{25} \]

F) When a term contains both a number and a letter, raise each part correctly. For example,

\[ (4m^2)^3 = 4^3 \times (m^2)^3 = 64m^6 \]

Be careful not to add the powers here: when raising a power to a power, you multiply the indices.

G) After expanding powers, simplify the fraction by treating numbers and letters separately. For example,

\[\begin{aligned} \frac{(6t^2)^2}{(3t^3)} &= \frac{36t^4}{3t^3} \\ &= 12t^{4-3} \\ &= 12t \end{aligned}\]

This method helps you avoid mistakes and keeps the working clear.

H) Always check whether the simplified result really matches the statement given. For example, if a statement says an expression is equal to \(k\), but your simplification gives \(2k\) or \(1\), then the statement is false. A small difference means the statement is not correct.

I) Remember the restriction on division: you cannot divide by \(0\). So in algebraic fractions such as \(\frac{r^5}{r^5}\), the simplification to \(1\) is only valid when \(r eq 0\).