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Indices brackets and combinations
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Indices: Brackets and Combinations

Indices: Brackets and Combinations

Indices: Brackets and Combinations

We can raise an index to another index, and combine with other rules such as multiplication and division.

1

Sometimes we need to raise a power to a power. For example:

(x3)2(x^3)^2

2

Here, we are squaring x3x^3, what is another way of writing (x3)2(x^3)^2?

3

We now have a multiplication, what is x3×x3x^3 \times x^3?

4

We have now found our answer!

(x3)2=x6(x^3)^2=x^6

5

Now we know that (x3)2=x6(x^3)^2=x^6, how are the powers related?

1

We have found our rule!

When raising a power to a power, shown by brackets, multiply the powers.

2

We can use the rule to simplify (42)4(4^2)^4 as a single power of 44.

(42)4=42×4=48(4^2)^4=4^{2 \times4}=4^8

3

Try this one, what is (65)3(6^5)^3 as a power of 6?

What is (1112)9(11^{12})^9 as a power of 1111?

What happens if the power is 0? Like in303^0?

1

Indices follow a pattern...

Let's use powers of 33 to identify this pattern.

2

What is 333^3

3

We've found 333^3, dividing this by 33 gives 333=32\dfrac{3^3}{3}=3^2. What is 323^2?

4

Similarly 323=31\dfrac{3^2}{3}=3^1. What is 313^1?

5

Each time the power is reduced by 11, we have divided by 33.

333=32\dfrac{3^3}{3}=3^2 and 323=31\dfrac{3^2}{3}=3^1

6

Continuing that pattern, what do we get in terms of powers of 33 when we divide 313\dfrac{3^1}{3}?

7

Okay, so what does 303^0 equal?

To find 303^0, we need to calculate 3÷33 \div 3

8

What is 3÷33\div3?

9

30=13^0 = 1

This is the same for any number to the power of 0.

What is the value of 12012^0?

Let's combine these rules and simplify (x6)2x3×x4\dfrac{(x^6)^2}{x^3 \times x^4}

1

What are we left with when we simplify the numerator (x6)2(x^6)^2?

2

Now simplify the denominator x3×x4x^3 \times x^4

3

Our expression can be rewritten as x12x7\dfrac{x^{12}}{x^{7}}

By subtracting the index numbers (since it is a division), we can find the simplest version.

4

Simplify x12x7\dfrac{x^{12}}{x^{7}}

Let's try another:

(x13x5)4\bigg(\dfrac{x^{13}}{x^5}\bigg)^4

1

Divide the powers inside the bracket

2

Now raise to the power of 44

3

Nice! 👍

The final answer is x32x^{32}.

1

Summary! When raising a power to a power, shown by using brackets, multiply the powers.

(x3)4=x12(x^3)^4=x^{12}

2

Any number raised to the power 00 is 11.

120=70=112^0=7^0=1

3

The rules can be combined to simplify complex expressions. All the rules depend on the base number being the same.

(x7)2x7×x9=x14x16=x−2\dfrac{(x^7)^2}{x^7\times x^9}=\dfrac{x^{14}}{x^{16}}=x^{-2}