Explore: Equations with Fractional Exponents Solutions

Equations with Fractional Exponents Solutions

Radical equations can be written using fractional exponents .
Use the reciprocal of a fractional exponent to isolate the variable.
If the numerator, n, of a fractional exponent is an even number, then: $(x^{\frac{n}{d}})^{\frac{d}{n}} = |x|$
Bases with even powers will only have positive answers.

Example 4

Solve.
$\frac{2}{5} {(5 x - 8)}^{\frac{1}{3}} + 3 = 5$

Implement
$\begin{array}{rcl} \frac{2}{5} {(5 x - 8)}^{\frac{1}{3}} & = & 2 \\ {(5 x - 8)}^{\frac{1}{3}} = 5 {({(5 x - 8)}^{\frac{1}{3}})}^{3} = {(5)}^{3} \end{array}$
$\begin{array}{rcl} 5 x - 8 & = & 125 \\ 5 x & = & 133 \\ x & = & \frac{133}{5} \end{array}$

Check
$\begin{array}{rcl} \frac{2}{5} {(5 (\frac{133}{5}) - 8)}^{\frac{1}{3}} + 3 & = & 5 \\ \frac{2}{5} {(133 - 8)}^{\frac{1}{3}} + 3 & = & 5 \\ \frac{2}{5} {(125)}^{\frac{1}{3}} + 3 & = & 5 ✓ \end{array}$

Explain

Isolate the expression with the rational exponent
Raise both sides to the reciprocal of the exponent
Isolate the variable

Example 5

Solve.
${(2 x + 5)}^{\frac{1}{2}} = x + 3$

$\begin{array}{rcl} {({(2 x + 5)}^{\frac{1}{2}})}^{2} & = & {(x + 3)}^{2} \\ 2 x + 5 & = & x^{2} + 6 x + 9 \\ 0 & = & x^{2} + 4 x + 4 \\ (x + 2) (x + 2) & = & 0 \\ x & = & - 2 \end{array}$

Check

$\begin{array}{rcl} {(2 (- 2) + 5)}^{\frac{1}{2}} & = & (- 2) + 3 \\ {(1)}^{\frac{1}{2}} & = & 1 ✓ \end{array}$

Example 6

Solve.
${(3 x + 1)}^{\frac{2}{3}} = 16$

Implement

${({(3 x + 1)}^{\frac{2}{3}})}^{\frac{3}{2}} = {(16)}^{\frac{3}{2}} {(16)}^{\frac{3}{2}} = {(16^{\frac{1}{2}})}^{3} = 4^{3} = 64$
$|3 x + 1| = 64 \begin{matrix} Case 1 & Case 2 \\ 3 x + 1 = 64 or & - (3 x + 1) = 64 \\ 3 x = 63 & 3 x + 1 = - 64 \\ x = 21 & 3 x = - 65 \\ x = - \frac{65}{3} \\ x = - \frac{65}{3}, 21 \end{matrix}$

Explain

$(x^{\frac{n}{d}})^{\frac{d}{n}} = |x|$ when n is an even number
Solve Case 1 and Case 2

Check

$\begin{matrix} {(3 (21) + 1)}^{\frac{2}{3}} = 16 & {(3 (- \frac{65}{3}) + 1)}^{\frac{2}{3}} = 16 \\ {(63 + 1)}^{\frac{2}{3}} = 16 & {(- 65 + 1)}^{\frac{2}{3}} = 16 \\ {(64)}^{\frac{2}{3}} = 16 & {(- 64)}^{\frac{2}{3}} = 16 \\ {(\sqrt[3]{64})}^{2} = 16 & {(\sqrt[3]{- 64})}^{2} = 16 \\ {(4)}^{2} = 16 ✓ & {(- 4)}^{2} = 16 ✓ \end{matrix}$

A negative number raised to an even power is positive.

Note

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