Indices (OCR GCSE Maths): Revision Notes

Indices

What are Indices?

Indices (also known as exponents or powers) are the small numbers or letters that float next to a number or letter, indicating how many times to multiply the base by itself.

Important Points about Indices

1. Indices apply only to the number or letter directly to their right:

• In $a^2bc$, only the $a$ is squared. The power does not apply to $b$ or $c$.

2. Indices do not mean multiply:

• $6^4$ means $6×6×6×6$, not $6×4.$

Rule 1 – The Multiplication Rule

Using Fancy Notation:

$$a^m×a^n=a^{m+n}$$

What it Actually Means:

• When multiplying two terms with the same base, you add the powers.

If There Are Numbers in Front of the Bases:

• You multiply the numbers as usual and then add the powers.

Remember:

• If a base does not appear to have a power, it actually has a disguised $1$.
  • For instance, $2ab^2c=2a^1b^2c^1$.

Common Mistakes to Avoid

• Wrong: $x^3×x^4=x^{12}$ (This is incorrect because you add the powers, not multiply them.)
  • Correct: $x^3×x^4=x^7$

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Rule 2 – The Division Rule

Using Fancy Notation:

$$\frac{a^m}{a^n} = a^{m-n} \quad \text{or} \quad a^m \div a^n = a^{m-n}$$

What it Actually Means:

• Whenever you are dividing two terms with the same base, you subtract the powers.

If There Are Numbers in Front of the Bases:

• Divide those numbers as usual and then subtract the powers.

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Rule 3 – The Power of a Power Rule

Using Fancy Notation:

$$\left( a^m \right)^n = a^{m \times n}$$

What it Actually Means:

• Whenever you have a base raised to a power and it's raised again to another power, you multiply the powers together while keeping the base the same.

If There Are Numbers in Front of the Bases:

• Raise the number to the power first and then handle the base's power.

Common Mistakes to Avoid

• Division Rule:
  • Wrong: $x^{12} \div x^4 = x^3$ (This is incorrect because the correct operation is subtraction, not division of powers.)
  • Correct: $x^{12} \div x^4 = x^8$
• Power of a Power Rule:
  • Wrong: $(x^3)^5 = x^8$ (This is incorrect because you should multiply the powers, not add them.)
  • Correct: $(x^3)^5 = x^{15}$

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Applying All Three Rules

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Rule 4: The Zero Index

Fancy Notation:

$$a^0 = 1$$

What It Actually Means:

Anything raised to the power of zero is $1$.

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Rule 5: Negative Indices

Fancy Notation:

$$a^{-m} = \frac{1}{a^m}$$

What It Actually Means:

A negative sign in front of a power is the same as writing "one divided by the base and power." The posh term for this is the reciprocal.

Watch Out:

Only the power and base are flipped over, nothing else!

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Rule 6: Fractional Indices

Fancy Notation:

$$a^{\frac{1}{n}} = \sqrt[n]{a}$$

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What It Actually Means:

When a power is a fraction, it means you take the root of the base, and which root you take depends on the number on the bottom of the fraction!

• The power of $\frac{1}2$ means take the square root:

$$a^{\frac{1}{2}} = \sqrt{a}$$

• The power of $\frac{1}3$ means take the cube root:

$$a^{\frac{1}{3}} = \sqrt[3]{a}$$

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Step 1: Flip It

Step 2: Root It

Step 3: Power It

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Step 1: Flip It

If there is a negative sign in front of your power, flip the base over, which will turn the power positive.

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Step 2: Root It

If your power is a fraction, deal with the denominator (the bottom of the fraction) by rooting your base.

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Step 3: Power It

Once you've done the first two steps, raise your base to the remaining power (the numerator).

Worked Examples

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