Index Notation 2
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Question 1 of 5
1. Question
Simplify`2a xx 3b xx a xx a`Hint
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Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$First, separate the numbers and the variables.`2a xx 3b xx a xx a` `=` `2 xx a xx 3 xx b xx a xx a` Then, bring like terms together.`2 xx a xx 3 xx b xx a xx a` `=` `2 xx 3 xx a xx a xx a xx b` Next, using the Product of Powers, bring similar bases together.Remember that any base without a power means to the power of `1`.`2 xx 3 xx a xx a xx a xx b` `=` `2^1``xx``3^1``xx``a^1``xx``a^1``xx``a^1``xx``b^1` `=` $$\color{#00880A}{6} \times \color{#9a00c7}{a}^{1+1+1} \times \color{#e65021}{b}^1$$ `=` $$\color{#00880A}{6} \times \color{#9a00c7}{a}^3 \times \color{#e65021}{b}$$ `=` `6a^3b` `6a^3b` -
Question 2 of 5
2. Question
Simplify`5c xx d xx d xx 4d xx c xx d`Hint
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Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$First, separate the numbers and the variables.`5c xx d xx d xx 4d xx c xx d` `=` `5 xx c xx d xx d xx 4 xx d xx c xx d` Then, bring like terms together.`5 xx c xx d xx d xx 4 xx d xx c xx d` `=` `5 xx 4 xx c xx c xx d xx d xx d xx d` Next, using the Product of Powers, bring similar bases together.Remember that any base without a power means to the power of `1`.`5 xx 4 xx c xx c xx d xx d xx d xx d` `=` `5^1``xx``4^1``xx``c^1``xx``c^1``xx``d^1``xx``d^1``xx``d^1``xx``d^1` `=` $$\color{#00880A}{20} \times \color{#9a00c7}{c}^{1+1} \times \color{#e65021}{d}^{1+1+1+1}$$ `=` $$\color{#00880A}{20} \times \color{#9a00c7}{c}^2 \times \color{#e65021}{d}^4$$ `=` `20c^2 d^4` `20c^2 d^4` -
Question 3 of 5
3. Question
Simplify`y xx y^5 xx y^3 xx y`Hint
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Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$Notice the bases are the same `(y)`.Since we are multiplying the bases, add the powers.Any base without a power means to the power of `1`.`y xx y^5 xx y^3 xx y` `=` $$\color{#00880A}{y}^1 \times \color{#00880A}{y}^5 \times \color{#00880A}{y}^3 \times \color{#00880A}{y}^1$$ `=` $$\color{#00880A}{y}^{1+5+3+1}$$ Use the Product of Powers `=` `y^10` `y^10` -
Question 4 of 5
4. Question
Simplify`4m^2 xx 3m xx m xx m^3`Hint
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Exceptional!
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Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$First, separate the numbers and the variables and bring like terms together.`4m^2 xx 3m xx m xx m^3` `=` `4 xx m^2 xx 3 xx m xx m xx m^3` `=` `4 xx 3 xx m^2 xx m xx m xx m^3` Next, using the Product of Powers, bring similar bases together.Remember that any base without a power means to the power of `1`.`4 xx 3 xx m^2 xx m xx m xx m^3` `=` `4^1``xx``3^1``xx``m^2``xx``m^1``xx``m^1``xx``m^3` `=` $$\color{#00880A}{12} \times \color{#9a00c7}{m}^{2+1+1+3}$$ `=` $$\color{#00880A}{12} \times \color{#9a00c7}{m}^7$$ `=` `12m^7` `12m^7` -
Question 5 of 5
5. Question
Simplify`(m xx m xx m xx m xx n xx n)/(m xx n)`Hint
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Product of Powers
$${\color{#00880A}{a}^m}\times{\color{#00880A}{a}^n}=\color{#00880A}{a}^{m+n}$$Quotient of Powers
$${\color{#00880A}{a}^m}\div{\color{#00880A}{a}^n}=\frac{{\color{#00880A}{a}^m}}{{\color{#00880A}{a}^n}}=\color{#00880A}{a}^{m-n}$$Using the Product of Powers, bring similar bases together.Remember that any base without a power means to the power of `1`.`(m xx m xx m xx m xx n xx n)/(m xx n)` `=` $$\frac{\color{#00880A}{m}^1 \times \color{#00880A}{m}^1 \times \color{#00880A}{m}^1 \times \color{#00880A}{m}^1 \times \color{#9a00c7}{n}^1 \times \color{#9a00c7}{n}^1}{m^1 \times n^1}$$ `=` $$\frac{\color{#00880A}{m}^{1+1+1+1} \times \color{#9a00c7}{n}^{1+1}}{m^1 \times n^1}$$ `=` $$\frac{\color{#00880A}{m}^4 \times \color{#9a00c7}{n}^2}{m^1 \times n^1}$$ Simplify further using the Quotient of Powers.$$\frac{\color{#00880A}{m}^4 \times \color{#9a00c7}{n}^2}{\color{#00880A}{m}^1 \times \color{#9a00c7}{n}^1}$$ $$\frac{\color{#00880A}{m}^4}{\color{#00880A}{m}^1} \times \frac{\color{#9a00c7}{n}^2}{\color{#9a00c7}{n}^1}$$ `=` $$\color{#00880A}{m}^{4-1} \times \color{#9a00c7}{n}^{2-1}$$ `=` $$\color{#00880A}{m}^3 \times \color{#9a00c7}{n}^1$$ `=` `m^3 n` A power of `1` does not need to be written `m^3 n`
Quizzes
- Index Notation 1
- Index Notation 2
- Index Notation 3
- Multiply Indices 1
- Multiply Indices 2
- Multiply Indices 3
- Multiply Indices 4
- Divide Indices 1
- Divide Indices 2
- Powers of a Power 1
- Powers of a Power 2
- Powers of a Power 3
- Powers of a Power 4
- Zero Powers 1
- Zero Powers 2
- Negative Indices 1
- Negative Indices 2
- Negative Indices 3
- Fractional Indices 1
- Fractional Indices 2
- Fractional Indices 3
- Mixed Operations with Indices 1
- Mixed Operations with Indices 2