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.

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For example: $5^4$ means $5×5×5×5.$

Important Points about Indices

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 $c$ .

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.

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Example:

x^3×x^4=x^{3+4}=x^7

If There Are Numbers in Front of the Bases:

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

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Example:

2a^3b^2×5a^2b^3=(2×5)×a^{3+2}×b^{2+3}=10a^5b^5

Remember:

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

Common Mistakes to Avoid

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

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.

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Example:

x^{12} \div x^4 = x^{12-4} = x^8

If There Are Numbers in Front of the Bases:

Divide those numbers as usual and then subtract the powers.

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Example:

\frac{20k^{10}}{5k^5} = \frac{20}{5} \times k^{10-5} = 4k^5

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.

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Example:

(x^3)^5 = x^{3 \times 5} = x^{15}

If There Are Numbers in Front of the Bases:

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

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Example:

(2a^3)^5 = 2^5 \times a^{3 \times 5} = 32a^{15}

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: $(x^3)^5 = x^{15}$

Applying All Three Rules

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Example 1: Simplifying the Expression

Apply the Division Rule (Rule $2$ ):

\frac{x^{11}}{x^5} = x^{11 - 5} = x^6

Final Answer:

x^6

Apply the Multiplication Rule (Rule $1$ ):

x^3 \times x^8 = x^{3 + 8} = x^{11}

Now the expression is:

\frac{x^{11}}{x^5}

Apply the Power of a Power Rule (Rule $3$ ):

(x^2)^4 = x^{2 \times 4} = x^8

Now the expression is:

\frac{x^3 \times x^8}{x^5}

Step-by-Step Solution:

\frac{x^3 \times (x^2)^4}{x^5}

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Example 2: Simplifying the Expression

\frac{(5^3)^2 \times (5^2)^{10}}{(5^5)^2}

Step-by-Step Solution:

Apply the Power of a Power Rule (Rule $3$ ) to each part:

(5^3)^2 = 5^{3 \times 2} = 5^6

(5^2)^{10} = 5^{2 \times 10} = 5^{20}

(5^5)^2 = 5^{5 \times 2} = 5^{10}

Now the expression is:

\frac{5^6 \times 5^{20}}{5^{10}}

Apply the Multiplication Rule (Rule $1$ ):

5^6 \times 5^{20} = 5^{6 + 20} = 5^{26}

Now the expression is:

\frac{5^{26}}{5^{10}}

Apply the Division Rule (Rule $2$ ):

\frac{5^{26}}{5^{10}} = 5^{26 - 10} = 5^{16}

Final Answer:

5^{16}

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Example 3: Simplifying the Expression

\frac{(5v^4)^2 \times (2v^5)^4}{50v}

Step-by-Step Solution:

Apply the Power of a Power Rule (Rule $3$ ):

(5v^4)^2 = 5^2 \times v^{4 \times 2} = 25v^8 \\

(2v^5)^4 = 2^4 \times v^{5 \times 4} = 16v^{20}

Now the expression is:

\frac{25v^8 \times 16v^{20}}{50v}

Apply the Multiplication Rule (Rule $2$ ):

25v^8 \times 16v^{20} = 400v^{8+20} = 400v^{28}

Now the expression is:

\frac{400v^{28}}{50v}

Apply the Division Rule (Rule $2$ ):

\frac{400v^{28}}{50v} = \frac{400}{50} \times v^{28-1} = 8v^{27}

Final Answer:

8v^{27}

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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Examples:

$x^0 = 1 \\$
$17^0 = 1 \\$
$5x^0 = 5 \times 1 = 5$

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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Examples:

$x^{-2} = \frac{1}{x^2} \\$
$5^{-4} = \frac{1}{5^4} \\$
$5a^{-2} = 5 \times \frac{1}{a^2} = \frac{5}{a^2} \\$
$(3^{-1})^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \\$
$(\frac{3}2)^{-3} = \frac{2^3}{3^3} = \frac{8}{27}

Rule 6: Fractional Indices

Fancy Notation:

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

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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Examples:

64^{\frac{1}{2}} = \sqrt{64} = 8 \\

27^{\frac{1}{3}} = \sqrt[3]{27} = 3 \quad \text{because} \quad 3 \times 3 \times 3 = 27 \\

32^{\frac{5}{5}} = 32 \quad \text{(simplifies to 32 because the powers cancel out)}

Tip: For fractional indices, it's worth learning your powers of $2$ and $3$ :

$2^2=4$
$3^2=9$
$2^3=8$
$3^3=27$
$2^4=16$
$3^4=81$
$2^5=32$
$2^6=64$

Step 1: Flip It

Step 2: Root It

Step 3: Power It

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.

Step 2: Root It

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

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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Example 1:

8^{−23}

Flip It:

(\frac{1}8)^{\frac23}

Root It:

\left(\sqrt[3]{\frac{1}{8}}\right)^2 = \left(\frac{1}{2}\right)^2

Power It:

\frac{1}{4}

So, \ 8^{-\frac{2}{3}} = \frac{1}{4}

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Example 2:

\left(\frac{1}{64}\right)^{-\frac{5}{6}}

Flip It:

64^{\frac{5}{6}}

Root It:

\left(\sqrt[6]{64}\right)^5 = \left(\sqrt[6]{2^6}\right)^5 = 2^5

Power It: $32$

So, \left(\frac{1}{64}\right)^{-\frac{5}{6}} = 32

Indices Simplified Revision Notes for GCSE OCR Maths

Indices

What are Indices?

Important Points about Indices

Rule 1 – The Multiplication Rule

Common Mistakes to Avoid

Rule 2 – The Division Rule

Rule 3 – The Power of a Power Rule

Common Mistakes to Avoid

Applying All Three Rules

Example 1: Simplifying the Expression

Rule 4: The Zero Index

Rule 5: Negative Indices

Rule 6: Fractional Indices

Step 1: Flip It

Step 2: Root It

Step 3: Power It

Step 1: Flip It

Step 2: Root It

Step 3: Power It

Worked Examples

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