Indices (B) Worksheet | Printable PDF Worksheets - Free Printable

Here is the complete solution to the worksheet “Indices (B)”, broken down by section with explanations.

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## Section A: Find the values.

1) \(3^0\)

Any non-zero number raised to the power of 0 is 1.
✔ Answer: 1

2) \((-2)^3\)

This means \(-2 \times -2 \times -2 = 4 \times -2 = -8\).
✔ Answer: -8

3) \((-1)^{20}\)

Since 20 is even, a negative base raised to an even power becomes positive.
\((-1)^{20} = 1\)
✔ Answer: 1

4) \(7^3\)

\(7 \times 7 = 49\), then \(49 \times 7 = 343\)
✔ Answer: 343

5) \((-5)^3 + (-4)^2\)

First, calculate each part:
- \((-5)^3 = -125\)
- \((-4)^2 = 16\)
Then add: \(-125 + 16 = -109\)
✔ Answer: -109

6) \(\left(\frac{3}{4}\right)^2\)

Square numerator and denominator: \(\frac{3^2}{4^2} = \frac{9}{16}\)
✔ Answer: \(\frac{9}{16}\)

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## Section B: Simplify.

We use the laws of indices:

- \(a^m \times a^n = a^{m+n}\)
- \(a^m \div a^n = a^{m-n}\)
- \(a^0 = 1\) (for \(a \neq 0\))

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1) \(5^2 \times 5^4 = 5^{2+4} = 5^6\) ✔ *(already given)*

2) \(9^{11} \times 9^2 = 9^{11+2} = 9^{13}\)

✔ Answer: \(9^{13}\)

3) \(2^5 \times 2 = 2^5 \times 2^1 = 2^{5+1} = 2^6\)

✔ Answer: \(2^6\)

4) \(8^5 \times 8^0 = 8^5 \times 1 = 8^5\)

✔ Answer: \(8^5\)

5) \(7^5 \div 7^3 = 7^{5-3} = 7^2\)

✔ Answer: \(7^2\)

6) \(12^{10} \div 12^6 = 12^{10-6} = 12^4\)

✔ Answer: \(12^4\)

7) \(4^{21} \div 4 = 4^{21} \div 4^1 = 4^{21-1} = 4^{20}\)

✔ Answer: \(4^{20}\)

8) \(5^3 \div 5^3 = 5^{3-3} = 5^0 = 1\)

✔ Answer: 1

9) \(2^{-3} \times 2^8 = 2^{-3+8} = 2^5\)

✔ Answer: \(2^5\)

10) \(6^4 \div 6^{12} = 6^{4-12} = 6^{-8}\)

✔ Answer: \(6^{-8}\)

11) \(15^2 \times 15^{-9} = 15^{2-9} = 15^{-7}\)

✔ Answer: \(15^{-7}\)

12) \(7^{10} \div 7^{14} \times 7 = 7^{10-14} \times 7^1 = 7^{-4} \times 7^1 = 7^{-3}\)

✔ Answer: \(7^{-3}\)

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## Section C: Simplify.

Use the laws:
- \(\frac{a^m}{a^n} = a^{m-n}\)
- \(a^m \times a^n = a^{m+n}\)

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1) \(\frac{2^{10}}{2^4} = 2^{10-4} = 2^6\)

✔ Answer: \(2^6\)

2) \(\frac{8^{-1}}{8^2} = 8^{-1-2} = 8^{-3}\)

✔ Answer: \(8^{-3}\)

3) \(\frac{10^{-12}}{10^7} = 10^{-12-7} = 10^{-19}\)

✔ Answer: \(10^{-19}\)

4) \(\frac{3^{-6}}{3^{-1}} = 3^{-6 - (-1)} = 3^{-5}\)

✔ Answer: \(3^{-5}\)

5) \(\frac{3^3 \times 3^5}{3^2} = \frac{3^{8}}{3^2} = 3^{8-2} = 3^6\)

✔ Answer: \(3^6\)

6) \(\frac{6^{-2} \times 6^4}{6^{-8}} = \frac{6^{2}}{6^{-8}} = 6^{2 - (-8)} = 6^{10}\)

✔ Answer: \(6^{10}\)

7) \(\frac{7^{-8} \times 7^{-7}}{7^{-4}} = \frac{7^{-15}}{7^{-4}} = 7^{-15 - (-4)} = 7^{-11}\)

✔ Answer: \(7^{-11}\)

8) \(\frac{14^{-3}}{14^{-11} \times 14^{-19}} = \frac{14^{-3}}{14^{-30}} = 14^{-3 - (-30)} = 14^{27}\)

✔ Answer: \(14^{27}\)

9) \(\frac{4^8 \times 4^5}{4^2 \times 4^4} = \frac{4^{13}}{4^{6}} = 4^{13-6} = 4^7\)

✔ Answer: \(4^7\)

10) \(\frac{6^{-7} \times 6^3}{6^{-1} \times 6} = \frac{6^{-4}}{6^{0}} = 6^{-4} \div 1 = 6^{-4}\)

✔ Answer: \(6^{-4}\)

11) \(\frac{20^{-17} \times 20^9}{20^4 \times 20^{-11}} = \frac{20^{-8}}{20^{-7}} = 20^{-8 - (-7)} = 20^{-1}\)

✔ Answer: \(20^{-1}\)

12) \(\frac{5e^8 \times 4e^5}{2e^2 \times 10e^4}\)

First, simplify coefficients and powers separately.

Numerator: \(5 \times 4 \times e^{8+5} = 20e^{13}\)

Denominator: \(2 \times 10 \times e^{2+4} = 20e^6\)

So: \(\frac{20e^{13}}{20e^6} = 1 \cdot e^{13-6} = e^7\)

✔ Answer: \(e^7\)

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## Extension: Investigate and Simplify

The pattern shown is:
\((a^m)^n = a^{m \times n}\)

Left Column — Investigate:

- \((2^1)^2 = 2^{1 \times 2} = 2^2\)
- \((2^2)^2 = 2^{2 \times 2} = 2^4\)
- \((2^3)^2 = 2^{3 \times 2} = 2^6\)
- \((2^4)^2 = 2^{4 \times 2} = 2^8\)
- \((2^5)^2 = 2^{5 \times 2} = 2^{10}\)

✔ Answers:
\(2^4, 2^6, 2^8, 2^{10}\)

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Middle Column — Investigate:

- \((5^1)^3 = 5^{1 \times 3} = 5^3\)
- \((5^2)^3 = 5^{2 \times 3} = 5^6\)
- \((5^3)^3 = 5^{3 \times 3} = 5^9\)
- \((5^4)^3 = 5^{4 \times 3} = 5^{12}\)
- \((5^5)^3 = 5^{5 \times 3} = 5^{15}\)

✔ Answers:
\(5^6, 5^9, 5^{12}, 5^{15}\)

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Right Column — Use results to simplify:

Apply \((a^m)^n = a^{m \times n}\)

- \((4^2)^4 = 4^{2 \times 4} = 4^8\)
- \((6^3)^8 = 6^{3 \times 8} = 6^{24}\)
- \((9^{-5})^6 = 9^{-5 \times 6} = 9^{-30}\)
- \((11^{-2})^{-7} = 11^{-2 \times -7} = 11^{14}\)

✔ Answers:
\(4^8, 6^{24}, 9^{-30}, 11^{14}\)

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## ✔ Final Summary of All Answers:

Section A:

1) 1
2) -8
3) 1
4) 343
5) -109
6) \(\frac{9}{16}\)

Section B:

1) \(5^6\)
2) \(9^{13}\)
3) \(2^6\)
4) \(8^5\)
5) \(7^2\)
6) \(12^4\)
7) \(4^{20}\)
8) 1
9) \(2^5\)
10) \(6^{-8}\)
11) \(15^{-7}\)
12) \(7^{-3}\)

Section C:

1) \(2^6\)
2) \(8^{-3}\)
3) \(10^{-19}\)
4) \(3^{-5}\)
5) \(3^6\)
6) \(6^{10}\)
7) \(7^{-11}\)
8) \(14^{27}\)
9) \(4^7\)
10) \(6^{-4}\)
11) \(20^{-1}\)
12) \(e^7\)

Extension:

Investigate:
- \((2^2)^2 = 2^4\), \((2^3)^2 = 2^6\), \((2^4)^2 = 2^8\), \((2^5)^2 = 2^{10}\)
- \((5^2)^3 = 5^6\), \((5^3)^3 = 5^9\), \((5^4)^3 = 5^{12}\), \((5^5)^3 = 5^{15}\)

Simplify:
- \((4^2)^4 = 4^8\)
- \((6^3)^8 = 6^{24}\)
- \((9^{-5})^6 = 9^{-30}\)
- \((11^{-2})^{-7} = 11^{14}\)

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