Sure, let's simplify these expressions using exponent laws and show all the steps.
### Part (a)
Simplify the expression:
[tex]\[ \frac{3 m^2 n^3 z}{6 m^{-1} n^2 z^2} \times \left(\frac{m^3 n}{2 z}\right)^0 \][/tex]
1. Simplify the fraction:
[tex]\[ \frac{3 m^2 n^3 z}{6 m^{-1} n^2 z^2} \][/tex]
- Divide the coefficients: [tex]\( \frac{3}{6} = \frac{1}{2} \)[/tex]
- Subtract the exponents for [tex]\( m \)[/tex]: [tex]\( m^{2 - (-1)} = m^{2 + 1} = m^3 \)[/tex]
- Subtract the exponents for [tex]\( n \)[/tex]: [tex]\( n^{3 - 2} = n^1 \)[/tex]
- Subtract the exponents for [tex]\( z \)[/tex]: [tex]\( z^{1 - 2} = z^{-1} \)[/tex]
So the simplified fraction becomes:
[tex]\[ \frac{1}{2} \times m^3 \times n^1 \times z^{-1} = \frac{1}{2} m^3 n z^{-1} \][/tex]
2. Simplify the term raised to the power of zero:
[tex]\[ \left(\frac{m^3 n}{2 z}\right)^0 = 1 \][/tex]
3. Multiply the results:
[tex]\[ \frac{1}{2} m^3 n z^{-1} \times 1 = \frac{1}{2} m^3 n z^{-1} \][/tex]
So, the simplified form is:
[tex]\[ \boxed{\frac{1}{2} m^3 n z^{-1}} \][/tex]
### Part (b)
Simplify the expression:
[tex]\[ \left(\frac{2 x^2}{7}\right) \times \left(\frac{y}{2 b^3}\right)^3 \][/tex]
1. Raise [tex]\( \frac{y}{2 b^3} \)[/tex] to the power of 3:
[tex]\[ \left(\frac{y}{2 b^3}\right)^3 = \frac{y^3}{(2 b^3)^3} = \frac{y^3}{2^3 b^{3 \times 3}} = \frac{y^3}{8 b^9} \][/tex]
2. Multiply the fractions:
[tex]\[ \left(\frac{2 x^2}{7}\right) \times \left(\frac{y^3}{8 b^9}\right) = \frac{2 x^2 y^3}{7 \times 8 b^9} = \frac{2 x^2 y^3}{56 b^9} \][/tex]
3. Simplify the coefficients:
[tex]\[ \frac{2 x^2 y^3}{56 b^9} = \frac{x^2 y^3}{28 b^9} \][/tex]
So, the simplified form is:
[tex]\[ \boxed{\frac{x^2 y^3}{28 b^9}} \][/tex]
### Part (c)
Simplify the expression:
[tex]\[ \left(\frac{x^{-7}}{x^{-9} y^{-10}}\right)^{-3} \][/tex]
1. Simplify the fraction inside the parentheses:
[tex]\[ \frac{x^{-7}}{x^{-9} y^{-10}} = x^{-7} \times \frac{1}{x^{-9} y^{-10}} = x^{-7} \times x^9 \times y^{10} = x^{-7 + 9} \times y^{10} = x^2 y^{10} \][/tex]
2. Raise the simplified expression to the power of [tex]\(-3\)[/tex]:
[tex]\[ \left(x^2 y^{10}\right)^{-3} = x^{2 \times (-3)} y^{10 \times (-3)} = x^{-6} y^{-30} \][/tex]
3. Express the result with positive exponents:
[tex]\[ x^{-6} y^{-30} = \frac{1}{x^6 y^{30}} \][/tex]
So, the simplified form is:
[tex]\[ \boxed{\frac{1}{x^6 y^{30}}} \][/tex]
That completes our step-by-step simplification!
