Roots and powers for AQA GCSE Maths
This page covers the following topics:
2. Simplifying roots
4. Simplifying powers
The root of a number, √, gives the number which multiplied by itself gives the required number. The root of a number can be estimated by finding the roots of integers above and below that number. The root of the desired number is between those two values and lies closer to the one which the initial integer is closest to. Numbers involving roots are called surds.
The laws of surds can be used to simplify expressions involving surds. Surds can be simplified by finding the factors of the number under the root which are square numbers and evaluating the roots of those numbers.
When the denominator of a fraction involves a surd, the fraction can be simplified by multiplying both the numerator and denominator by that surd. This process is called rationalising the denominator.
The power to which a value is raised describes the number of times the value is multiplied by itself. The power of a value can be estimated by finding the powers of the integers above and below the number. The power of the desired number will lie between the two.
Fractional powers are related to roots. The denominator of the fractional power is the root taken, whereas the numerator represents the power to which the value is raised. This can be used to interchange between fractional powers and roots.
When a number raised to a power is being raised to another power, the power rule may be used to simplify this. The power rule states that the base number is raised to the product of the two powers.
A negative index represents a reciprocal, meaning that a reciprocal of the value which is being raised to the negative power must be taken. When two terms with the same base number are multiplied together, they can be expressed as one term by adding their powers together. This is called the multiplication rule. In the case of division, the powers are subtracted instead. This is called the division rule.
Given that 25² × 5ˣ ÷ 125 = 625, find x.
25² × 5ˣ ÷ 125 = 625
25² × 5ˣ ÷ 125 = 5⁴
(5²)² × 5ˣ ÷ 5³ = 5⁴
5⁴ × 5ˣ ÷ 5³ = 5⁴ by the power rule
Using the multiplication and division rule, 4 + x − 3 = 4
1 + x = 4
x = 3
Calculate the value of 1.3 × 10².
1.3 × 10² = 1.3 × 10 × 10 = 130
Simplify the following expression: √12 ÷ √4 × √2.
√12 ÷ √4 × √2 = √(12 ÷ 4) × √2 = √3 × √2 = √(3 × 2) = √6
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