1.2 Exponents and scientific notation (Page 4/9)

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Try It

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

$\frac{{(−3 t)}^{2}}{{(−3 t)}^{8}}$
$\frac{f^{47}}{f^{49} \cdot f}$
$\frac{2 k^{4}}{5 k^{7}}$

$\frac{1}{{(−3 t)}^{6}}$
$\frac{1}{f^{3}}$
$\frac{2}{5 k^{3}}$

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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.

$b^{2} \cdot b^{−8}$
${(- x)}^{5} \cdot {(- x)}^{−5}$
$\frac{−7 z}{{(−7 z)}^{5}}$

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

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Try It

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

$t^{−11} \cdot t^{6}$
$\frac{25^{12}}{25^{13}}$

$t^{−5} = \frac{1}{t^{5}}$
$\frac{1}{25}$

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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 ${(p q)}^{3} .$ We begin by using the associative and commutative properties of multiplication to regroup the factors.

\begin{matrix} {(p q)}^{3} & = & \overset{3 factors}{\overset{}{(p q) \cdot (p q) \cdot (p q)}} \\ = & p \cdot q \cdot p \cdot q \cdot p \cdot q \\ = & \overset{3 factors}{\overset{}{p \cdot p \cdot p}} \cdot \overset{3 factors}{\overset{}{q \cdot q \cdot q}} \\ = & p^{3} \cdot q^{3} \end{matrix}

In other words, ${(p q)}^{3} = p^{3} \cdot q^{3} .$

A General Note

The power of a product rule of exponents

For any real numbers $a$ and $b$ and any integer $n,$ the power of a product rule of exponents states that

{(a b)}^{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.

${(a b^{2})}^{3}$
${(2 t)}^{15}$
${(−2 w^{3})}^{3}$
$\frac{1}{{(−7 z)}^{4}}$
${(e^{−2} f^{2})}^{7}$

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

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

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Try It

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

${(g^{2} h^{3})}^{5}$
${(5 t)}^{3}$
${(−3 y^{5})}^{3}$
$\frac{1}{{(a^{6} b^{7})}^{3}}$
${(r^{3} s^{−2})}^{4}$

$g^{10} h^{15}$
$125 t^{3}$
$−27 y^{15}$
$\frac{1}{a^{18} b^{21}}$
$\frac{r^{12}}{s^{8}}$

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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.

{(e^{−2} f^{2})}^{7} = \frac{f^{14}}{e^{14}}

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

\begin{matrix} {(e^{−2} f^{2})}^{7} & = & {(\frac{f^{2}}{e^{2}})}^{7} \\ = & \frac{f^{14}}{e^{14}} \end{matrix}

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{matrix} {(e^{- 2} f^{2})}^{7} & = & {(\frac{f^{2}}{e^{2}})}^{7} \\ = & \frac{{(f^{2})}^{7}}{{(e^{2})}^{7}} \\ = & \frac{f^{2 \cdot 7}}{e^{2 \cdot 7}} \\ = & \frac{f^{14}}{e^{14}} \end{matrix}

A General Note

The power of a quotient rule of exponents

For any real numbers $a$ and $b$ and any integer $n,$ the power of a quotient rule of exponents states that

{(\frac{a}{b})}^{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.

${(\frac{4}{z^{11}})}^{3}$
${(\frac{p}{q^{3}})}^{6}$
${(\frac{−1}{t^{2}})}^{27}$
${(j^{3} k^{−2})}^{4}$
${(m^{−2} n^{−2})}^{3}$

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

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Questions & Answers

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Ewa Reply

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Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity

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A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.

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A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike

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Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.

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the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon

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nuclei having the same Z and different N s

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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