Rewrite each expression so that each term is in the form kx^{n}, where k is a real number, x is a positive real number, and n is a rational number. a)x^{-frac{2}{3}} times x^{frac{1}{3}}=? b) frac{10x^{frac{1}{3}}}{2x^{2}}=? c) (3x^{frac{1}{4}})^{-2}=?

Question

Rewrite each expression so that each term is in the form

k x^{n},

where k is a real number, x is a positive real number, and n is a rational number.
a)

x^{- \frac{2}{3}} \times x^{\frac{1}{3}} = ?

b)

\frac{10 x^{\frac{1}{3}}}{2 x^{2}} = ?

c)

(3 x^{\frac{1}{4}})^{- 2} = ?

Answer 1

Step 1
We have to rewrite the given expression in the form of

k x^{n} .

Before going to simplify expression, first we need to know about rule of exponents.
1.

a^{m} \times a^{n} = a^{m + n}

2.

\frac{a^{m}}{a^{n}} = a^{m - n}

3.

(a^{m})^{n} = a^{(m \times n)}

Now, we will rewrite the given expression.
(a) Expression is

x^{- \frac{2}{3}} \times x^{\frac{1}{3}}

Given expression can be written as:

x^{- \frac{2}{3}} \times x^{\frac{1}{3}} = x^{(- \frac{2}{3} + \frac{1}{3})}

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

= x^{- \frac{1}{3}}

(b) Expression is

\frac{10 x^{\frac{1}{3}}}{2 x^{2}} .

Given expression can be written as:

\frac{10 x^{\frac{1}{3}}}{2 x^{2}} = \frac{10}{2} \times x^{\frac{1}{3}} \times x^{- 2}

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

= 5 x^{\frac{1 - 6}{3}}

= 5 x^{- \frac{5}{3}}

(c) Expression is

(3 x^{\frac{1}{4}})^{- 2}

Given expression can be written as:

(3 x^{\frac{1}{4}})^{- 2} = 3 x^{\frac{1}{4} \times - 2}

= 3 x^{- \frac{1}{2}}

Expert Answer