## What you need to know

Things to remember:

• Substitute (replace) the letter values with the number they are equal to.
• We use BIDMAS to evaluate the expression

Before we start, we just need to recall three bits of terminology:

• Evaluate means to find the value of, or to calculate, something.
• Non-linear means that there are powers and square roots.
• Expressions are mathematical phrases that include variables, numbers, and operators.

So, this topic is to do with finding the value of expressions that have powers and square roots. We will start with a linear first expression though.

Evaluate $5x+2$ when $x=3$?

This question is telling us that we need to find the value of $5x+2$ when $x=3$, so the first step in our 2 step program is to substitute this value in.

Step 1: Substitute the variable values into the expression.

$$5x+2=5\times3+2\text{ when } x=3$$

Step 2: Evaluate the expression using BIDMAS.

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Indices (Powers)

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Divisions

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Multiplications

$$5\times3+2=15+2$$

$$15+2=17$$

$$5x+2=17\text{ when } x=3$$

We follow these exact same steps when we have a non-linear expression.

Evaluate $3x^2-\sqrt{x}\div2$ when $x=4$?

Step 1: Substitute the variable value into the expression.

$$3x^2-\sqrt{x}\div2=3\times4^2-\sqrt{4}\div2\text{ when } x=4$$

Step 2: Evaluate the expression using BIDMAS.

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Indices (Powers)

$$3\times4^2-\sqrt{4}\div2=3\times16-2\div2$$

Note: Square rooting is actually a power of $\dfrac{1}{2}$, so we do them at the same time as whole number powers.

Divisions

$$3\times16-2\div2=3\times16-1$$

Multiplications

$$3\times16-1 =48-1$$

$$48-1=47$$

$$3x^2-\sqrt{x}\div2=47 \text{ when } x=16$$

## Example Questions

#### Question 1: Evaluate $x^3-x^2$ when $x=5$?

Step 1: Substitute the variable values into the expression.

$$x^3-x^2=5^3-5^2 \text{ when } x=5$$

Step 2: Evaluate the expression using BIDMAS.

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Indices (Powers)

$$5^3-5^2 = 125-25$$

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Multiplications

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$$125-25 = 100 \text{ when } x=5$$

$$x^3-x^2 = 100 \text{ when } x=5$$

#### Question 2: Evaluate $3x- \sqrt{x}\times4$ when $x=49$?

Step 1: Substitute the variable values into the expression.

$$3x- \sqrt{x}\times4 =3\times49- \sqrt{49}\times4 \text{ when } x=49$$

Step 2: Evaluate the expression using BIDMAS.

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Indices (Powers)

$$3\times49- \sqrt{49}\times4 =3\times49-7\times4$$

Divisions

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Multiplications

$$3\times49-7\times4=147-28$$

$$147-28=119$$
$$3x- \sqrt{x}\times4 = 119 \text{ when } x=49$$