Chapter 7.2, Problem 38PPE

(a).

To determine

To find: The reason for common denominator to simplify the given algebraic expression.

Explanation of Solution

Given information:

The given algebraic expression is $c^{\frac{3}{5}}\cdot c^{\frac{1}{2}}$ .

Interpretation:

The given algebraic expression is:

  $c^{\frac{3}{5}}\cdot c^{\frac{1}{2}}$

Here, the denominators of given numbers in the power are unequal. So, common denominator is required to simplify. To simplify. take the LCM (Least common multiple) of the denominators and express each of the given numbers with the LCM as the common denominator.

(b).

To determine

To find: The simplification of the given algebraic expression.

Answer to Problem 38PPE

The simplification of the given algebraic expression is $c^{\frac{11}{10}}$ .

Explanation of Solution

Given information:

The given algebraic expression is $c^{\frac{3}{5}}\cdot c^{\frac{1}{2}}$ .

Calculation:

The given algebraic expression is:

  $c^{\frac{3}{5}}\cdot c^{\frac{1}{2}}$

To determine the above value, take the LCM of the unequal denominators. Formula to calculate LCM of the two numbers $x$ and $y$ is:

  $LCM(x,y)=x\times y$

Substitute $5$ for $x$ and $2$ for $y$ in the above expression.

  $\begin{array}{c}LCM\left(5,2\right)=5\times 2\\ =10\end{array}$

Thus, the common denominator of the given numbers is $10$ .

To determine the value of $\frac{3}{5}$ in the common denominator, multiply the denominator and numerator of the given number by $2$ .

  $\begin{array}{c}\frac{3}{5}=\frac{3\times 2}{5\times 2}\\ =\frac{6}{10}\end{array}$

To determine the value of $\frac{1}{2}$ in the common denominator, multiply the denominator and numerator of the given number by $5$ .

  $\begin{array}{c}\frac{1}{2}=\frac{1\times 5}{2\times 5}\\ =\frac{5}{10}\end{array}$

Formula for the addition property of exponents with the same base is:

  $a^n\cdot a^m=a^{n+m}$

Substitute $c$ for $a$ , $\frac{3}{5}$ for $n$ and $\frac{1}{2}$ for $m$ in the above given expression.

  $c^{\frac{3}{5}}\cdot c^{\frac{1}{2}}=c^{\frac{3}{5}+\frac{1}{2}}$

The value whose sum is to be determined is:

  $\frac{3}{5}+\frac{1}{2}$

Substitute $\frac{6}{10}$ for $\frac{3}{5}$ and $\frac{5}{10}$ for $\frac{1}{2}$ in the above value.

  $\begin{array}{c}\frac{3}{5}+\frac{1}{2}=\frac{6}{10}+\frac{5}{10}\\ =\frac{6+5}{10}\\ =\frac{11}{10}\end{array}$

Substitute $\frac{11}{10}$ for $\frac{3}{5}+\frac{1}{2}$ in the above exponent expression.

  $c^{\frac{3}{5}}\cdot c^{\frac{1}{2}}=c^{\frac{11}{10}}$

Therefore, the simplification of the given algebraic expression is $c^{\frac{11}{10}}$ .