Explained: "Simplify each of the following. Express final results..."

High School
Algebra
Algebra I
Exponents and radicals

Anonym

Answered question

2021-10-27

Simplify each of the following. Express final results using positive exponents only. For example,

(2 x^{\frac{1}{2}}) (3 x^{\frac{1}{3}}) = 6 x^{\frac{5}{6}}

{(\frac{x^{2}}{y^{3}})}^{- \frac{1}{2}}

Answer & Explanation

Nicole Conner

Skilled2021-10-28Added 97 answers

We can add or multiply the exponents for simplification by using the formulas of exponents, such as
$a^{m} a^{n} = a^{m + n}$
${(a^{m})}^{n} = a^{m n}$
${(\frac{a^{m}}{b^{n}})}^{p} = \frac{a^{\mp}}{b^{n p}}$
$\frac{a^{m}}{b^{n}} = a^{m - n}$
Also we can use the expressions of exponents as
$\sqrt{a} = a^{\frac{1}{2}}$
$\sqrt{n} {a} = a^{\frac{1}{n}}$
$a^{- n} = \frac{1}{a^{n}}$
Consider the expression $(2 x^{\frac{1}{2}}) (3 x^{\frac{1}{3}}) = (6 x^{\frac{5}{6}}) \cdot {(\frac{x^{2}}{y^{3}})}^{\frac{- 1}{2}}$
In the left hand side we can use formula $a^{m} a^{n} = a^{m + n}$ , in the second term of right hand side can be simplify using the formula $a^{- n} = \frac{1}{a^{n}}$
thus we get
$(2 x^{\frac{1}{2}}) (3 x^{\frac{1}{3}}) = (6 x^{\frac{5}{6}}) {(\frac{x^{2}}{y^{3}})}^{\frac{- 1}{2}}$
$(2 \times 3) (x^{\frac{1}{2} + \frac{1}{3}}) = \frac{6 x^{\frac{5}{6}}}{{(\frac{x^{2}}{y^{3}})}^{\frac{- 1}{2}}}$
Now we can divide both sides by $6 x^{\frac{5}{6}}$ ${(\frac{x^{2}}{y^{3}})}^{\frac{1}{2}} = 1$
Hence we have the simplest form in positive exponents as $x = y^{\frac{3}{2}}$

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