Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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!
### 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!
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
