# 1.2 Exponents and scientific notation  (Page 4/9)

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Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

1. $\frac{{\left(-3t\right)}^{2}}{{\left(-3t\right)}^{8}}$
2. $\frac{{f}^{47}}{{f}^{49}\cdot f}$
3. $\frac{2{k}^{4}}{5{k}^{7}}$
1. $\frac{1}{{\left(-3t\right)}^{6}}$
2. $\frac{1}{{f}^{3}}$
3. $\frac{2}{5{k}^{3}}$

## Using the product and quotient rules

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

1. ${b}^{2}\cdot {b}^{-8}$
2. ${\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}$
3. $\frac{-7z}{{\left(-7z\right)}^{5}}$
1. ${b}^{2}\cdot {b}^{-8}={b}^{2-8}={b}^{-6}=\frac{1}{{b}^{6}}$
2. ${\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5-5}={\left(-x\right)}^{0}=1$
3. $\frac{-7z}{{\left(-7z\right)}^{5}}=\frac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1-5}={\left(-7z\right)}^{-4}=\frac{1}{{\left(-7z\right)}^{4}}$

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

1. ${t}^{-11}\cdot {t}^{6}$
2. $\frac{{25}^{12}}{{25}^{13}}$
1. ${t}^{-5}=\frac{1}{{t}^{5}}$
2. $\frac{1}{25}$

## Finding the power of a product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider $\text{\hspace{0.17em}}{\left(pq\right)}^{3}.\text{\hspace{0.17em}}$ We begin by using the associative and commutative properties of multiplication to regroup the factors.

In other words, $\text{\hspace{0.17em}}{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}.$

## The power of a product rule of exponents

For any real numbers $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and any integer $\text{\hspace{0.17em}}n,$ the power of a product rule of exponents states that

${\left(ab\right)}^{n}={a}^{n}{b}^{n}$

## Using the power of a product rule

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

1. ${\left(a{b}^{2}\right)}^{3}$
2. ${\left(2t\right)}^{15}$
3. ${\left(-2{w}^{3}\right)}^{3}$
4. $\frac{1}{{\left(-7z\right)}^{4}}$
5. ${\left({e}^{-2}{f}^{2}\right)}^{7}$

Use the product and quotient rules and the new definitions to simplify each expression.

1. ${\left(a{b}^{2}\right)}^{3}={\left(a\right)}^{3}\cdot {\left({b}^{2}\right)}^{3}={a}^{1\cdot 3}\cdot {b}^{2\cdot 3}={a}^{3}{b}^{6}$
2. ${\left(2t\right)}^{15}={\left(2\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}$
3. ${\left(-2{w}^{3}\right)}^{3}={\left(-2\right)}^{3}\cdot {\left({w}^{3}\right)}^{3}=-8\cdot {w}^{3\cdot 3}=-8{w}^{9}$
4. $\frac{1}{{\left(-7z\right)}^{4}}=\frac{1}{{\left(-7\right)}^{4}\cdot {\left(z\right)}^{4}}=\frac{1}{2,401{z}^{4}}$
5. ${\left({e}^{-2}{f}^{2}\right)}^{7}={\left({e}^{-2}\right)}^{7}\cdot {\left({f}^{2}\right)}^{7}={e}^{-2\cdot 7}\cdot {f}^{2\cdot 7}={e}^{-14}{f}^{14}=\frac{{f}^{14}}{{e}^{14}}$

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

1. ${\left({g}^{2}{h}^{3}\right)}^{5}$
2. ${\left(5t\right)}^{3}$
3. ${\left(-3{y}^{5}\right)}^{3}$
4. $\frac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}$
5. ${\left({r}^{3}{s}^{-2}\right)}^{4}$
1. ${g}^{10}{h}^{15}$
2. $125{t}^{3}$
3. $-27{y}^{15}$
4. $\frac{1}{{a}^{18}{b}^{21}}$
5. $\frac{{r}^{12}}{{s}^{8}}$

## Finding the power of a quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.

${\left({e}^{-2}{f}^{2}\right)}^{7}=\frac{{f}^{14}}{{e}^{14}}$

Let’s rewrite the original problem differently and look at the result.

$\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}$

It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

$\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{\left({f}^{2}\right)}^{7}}{{\left({e}^{2}\right)}^{7}}\hfill \\ & =& \frac{{f}^{2\cdot 7}}{{e}^{2\cdot 7}}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}$

## The power of a quotient rule of exponents

For any real numbers $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and any integer $\text{\hspace{0.17em}}n,$ the power of a quotient rule of exponents states that

${\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}$

## Using the power of a quotient rule

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

1. ${\left(\frac{4}{{z}^{11}}\right)}^{3}$
2. ${\left(\frac{p}{{q}^{3}}\right)}^{6}$
3. ${\left(\frac{-1}{{t}^{2}}\right)}^{27}$
4. ${\left({j}^{3}{k}^{-2}\right)}^{4}$
5. ${\left({m}^{-2}{n}^{-2}\right)}^{3}$
1. ${\left(\frac{4}{{z}^{11}}\right)}^{3}=\frac{{\left(4\right)}^{3}}{{\left({z}^{11}\right)}^{3}}=\frac{64}{{z}^{11\cdot 3}}=\frac{64}{{z}^{33}}$
2. ${\left(\frac{p}{{q}^{3}}\right)}^{6}=\frac{{\left(p\right)}^{6}}{{\left({q}^{3}\right)}^{6}}=\frac{{p}^{1\cdot 6}}{{q}^{3\cdot 6}}=\frac{{p}^{6}}{{q}^{18}}$
3. ${\left(\frac{-1}{{t}^{2}}\right)}^{27}=\frac{{\left(-1\right)}^{27}}{{\left({t}^{2}\right)}^{27}}=\frac{-1}{{t}^{2\cdot 27}}=\frac{-1}{{t}^{54}}=-\frac{1}{{t}^{54}}$
4. ${\left({j}^{3}{k}^{-2}\right)}^{4}={\left(\frac{{j}^{3}}{{k}^{2}}\right)}^{4}=\frac{{\left({j}^{3}\right)}^{4}}{{\left({k}^{2}\right)}^{4}}=\frac{{j}^{3\cdot 4}}{{k}^{2\cdot 4}}=\frac{{j}^{12}}{{k}^{8}}$
5. ${\left({m}^{-2}{n}^{-2}\right)}^{3}={\left(\frac{1}{{m}^{2}{n}^{2}}\right)}^{3}=\frac{{\left(1\right)}^{3}}{{\left({m}^{2}{n}^{2}\right)}^{3}}=\frac{1}{{\left({m}^{2}\right)}^{3}{\left({n}^{2}\right)}^{3}}=\frac{1}{{m}^{2\cdot 3}\cdot {n}^{2\cdot 3}}=\frac{1}{{m}^{6}{n}^{6}}$

#### Questions & Answers

what are you up to?
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
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Jeffrey
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Jeffrey
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Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
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Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
Hiiii
Chinni
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Ranjay
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ANSHU
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Chinni
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SORIE
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Abdel
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SORIE
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function