# Exponent problems – Grade 09 – set 01

In this post we will solve exponent questions with difficulty of grade 9 standard. All the questions are provided with solution.

To solve the questions, you should have basic understanding of. Click the red link to learn about the same.

Question 01
Simplify the below expressions;

\mathtt{( i) \ \frac{3xy^{2}\left( -2x^{2} y^{3}\right)}{6xy}}\\\ \\ \mathtt{( ii) \ \left( 9x^{-5} y^{2}\right)^{-3}}\\\ \\ \mathtt{( iii) \ \frac{1}{1+x^{a-b}} +\frac{1}{1+x^{b-a}}}\\\ \\ \mathtt{( iv) \ \frac{2^{n} \times 4^{n+1}}{2^{n-1} \times 4^{n-1}}}\\\ \\ \mathtt{( v) \ \sqrt{x^{5} y^{-2}}}

Solution

\mathtt{( i) \ \frac{3xy^{2}\left( -2x^{2} y^{3}\right)}{6xy}} \\\ \\
\mathtt{\Longrightarrow \ \frac{-6\times x^{2+1} \times y^{2+3}}{6xy}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-6x^{3} y^{5}}{6xy}}\\\ \\ \mathtt{\Longrightarrow -1x^{3-1} y^{5-1}}\\\ \\ \mathtt{\Longrightarrow \ -x^{2} y^{4}}

\mathtt{( ii) \ \left( 9x^{-5} y^{2}\right)^{-3}}\\\ \\ \mathtt{\Longrightarrow 9^{-3} \times x^{( -5\ \times -3)} \times y^{( 2\ \times -3)} \ }\\\ \\ \mathtt{\Longrightarrow \frac{1}{729} x^{15} y^{-6}}

\mathtt{( iii) \ \frac{1}{1+x^{a-b}} +\frac{1}{1+x^{b-a}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{x^{b-b} +x^{a-b}} +\frac{1}{x^{a-a} +x^{b-a}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{x^{-b}\left( x^{b} +x^{a}\right)} +\frac{1}{x^{-a}\left( x^{a} +x^{b}\right)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{\left( x^{a} +x^{b}\right)}\left(\frac{1}{x^{-b}} +\frac{1}{x^{-a}}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{\left( x^{a} +x^{b}\right)}\left( x^{b} +x^{a}\right)}\\\ \\ \mathtt{\Longrightarrow \ 1}

\mathtt{( iv) \ \frac{2^{n} \times 4^{n+1}}{2^{n-1} \times 4^{n-1}}}\\\ \\ \mathtt{\Longrightarrow \ 2^{n-( n-1)} \times 4^{n+1-( n-1)}}\\\ \\ \mathtt{\Longrightarrow \ 2^{1} \times 4^{2}}\\\ \\ \mathtt{\Longrightarrow \ 32}

\mathtt{( v) \ \sqrt{x^{5} y^{-2}}}\\\ \\ \mathtt{\Longrightarrow \ x^{\frac{5}{2}} \times \ y^{\frac{-2}{2}}}\\\ \\ \mathtt{\Longrightarrow \ x^{\frac{5}{2}} \times y^{-1}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x^{\frac{5}{2}}}{y}}

Question 02
If a = -3 and b = 2, then find the value of below expressions;

\mathtt{( i) \ a^{a} +b^{b}}\\\ \\ \mathtt{( ii) \ a^{b} +b^{a}}\\\ \\ \mathtt{( iii) \ ( a+b)^{ab}}

Solution
\mathtt{( i) \ a^{a} +b^{b}}\\\ \\

\mathtt{\Longrightarrow \ ( -3)^{-3} +\ ( 2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{( -3)^{3}} +\ 4}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1}{27} +4}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1+108}{27}}\\\ \\ \mathtt{\Longrightarrow \ \frac{107}{27}}

\mathtt{( ii) \ a^{b} +b^{a}} \\\ \\

\mathtt{\Longrightarrow \ ( -3)^{2} +( 2)^{-3}}\\\ \\ \mathtt{\Longrightarrow \ 9\ +\ \frac{1}{2^{3}}}\\\ \\ \mathtt{\Longrightarrow \ 9\ +\ \frac{1}{8}}\\\ \\ \mathtt{\Longrightarrow \ \frac{72+1}{8}}\\\ \\ \mathtt{\Longrightarrow \ \frac{73}{8}}

\mathtt{( iii) \ ( a+b)^{ab}} \\\ \\

\mathtt{\Longrightarrow \ ( -3+2)^{-3\times 2}}\\\ \\ \mathtt{\Longrightarrow \ ( -1)^{-6}}\\\ \\ \mathtt{\Longrightarrow \ 1}

Next Chapter : Exponents grade 09 problems – set 02