Question

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. (x - 4)3/2 = 27

Accepted Answer

Recognize that the equation involves a rational exponent: $$ (x - 4$$)^{\frac{3}$${2}} = 27 . $$The exponent $$\frac{3}{2}$$ means we are dealing with a power and a root combined. Rewrite the expression $$ (x - 4$$)^{\frac{3}$${2}}  as$$ $$ \left( (x - 4$$)^{\frac{1}$${2}} \$$right)^3$$  or $$equivalently $$ \left( \sqrt{x - 4} \$$right)^3$$  to $$better understand the operation. To isolate $$ x - 4 $$, raise both sides of the equation to the reciprocal power of $$ \frac{3}{2} $$, which is $$ \frac{2}{3} $$, so apply $$ \left( \cdot \$$right)^{\frac{2}$${3}}  to $$both sides:  $$ \left( (x - 4$$)^{\frac{3}$${2}} \$$right)^{\frac{2}$${3}} = 27$$^{\frac{2}$${3}} . $$Simplify the left side using the property $$ (a$$^{m}$$)^{n}$$ = $$a^{mn}$$ $$, so  (x$$ - $$4)^{\frac{3}$${2} 	imes \frac{2}{3}} = (x - $$4)^1 = x - 4$$ $$. Then calculate the right side $$ $$27^{\frac{2}$${3}} $$ by first finding the cube root of 27 and then squaring the result. After simplification, solve for  x$$ $ by adding 4 to both sides. Finally, check all proposed solutions by substituting them back into the original equation to ensure they satisfy it, especially because rational exponents can introduce extraneous solutions.

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. (x - 4)^3/2 = 27

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Concetti chiave

Rational Exponents

Isolating the Variable

Checking Solutions