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Answer :
Sure! Let's go through each expression and write out the step-by-step simplified forms for each of them.
### Simplifying the Expressions:
1. Expression 19: [tex]\( (3 x^{\frac{1}{2}})(4 x^{\frac{3}{3}}) \)[/tex]
- Simplify the exponents: [tex]\( x^{\frac{3}{3}} = x^1 = x \)[/tex].
- Combine powers: [tex]\( x^{\frac{1}{2}} \cdot x^1 = x^{\frac{1}{2} + 1} = x^{\frac{3}{2}} \)[/tex].
- Multiply the coefficients: [tex]\( 3 \cdot 4 = 12 \)[/tex].
- Final expression: [tex]\( 12x^{\frac{3}{2}} \)[/tex].
2. Expression 20: [tex]\( 2 b^{\frac{1}{2}}(3 b^{\frac{1}{2}} c^{\frac{1}{3}})^2 \)[/tex]
- Expand the bracket using power: [tex]\( (3 b^{\frac{1}{2}} c^{\frac{1}{3}})^2 = 9 b^1 c^{\frac{2}{3}} \)[/tex].
- Multiply with [tex]\( 2 b^{\frac{1}{2}} \)[/tex]:
- Combine the powers of [tex]\( b \)[/tex]: [tex]\( b^{\frac{1}{2} + 1} = b^{\frac{3}{2}} \)[/tex].
- Coefficients multiply: [tex]\( 2 \cdot 9 = 18 \)[/tex].
- Final expression: [tex]\( 18b^{\frac{3}{2}}c^{\frac{2}{3}} \)[/tex].
3. Expression 21: [tex]\( \left(x^{\frac{1}{2}} \cdot x^{\frac{3}{12}}\right)^4 \div x^{\frac{2}{3}} \)[/tex]
- Simplify inside the power: [tex]\( x^{\frac{1}{2} + \frac{1}{4}} = x^{\frac{3}{4}} \)[/tex].
- Apply the exponent: [tex]\( (x^{\frac{3}{4}})^4 = x^3 \)[/tex].
- Divide by [tex]\( x^{\frac{2}{3}} \)[/tex]: [tex]\( x^{3 - \frac{2}{3}} = x^{\frac{7}{3}} \)[/tex].
- Final expression: [tex]\( x^{\frac{7}{3}} \)[/tex].
4. Expression 22: [tex]\( \left(\frac{16 c^{14}}{81 d^{18}}\right)^{\frac{1}{2}} \)[/tex]
- Take the square root:
- [tex]\( \sqrt{16} = 4 \)[/tex]
- [tex]\( \sqrt{81} = 9 \)[/tex]
- [tex]\( \sqrt{c^{14}} = c^7 \)[/tex]
- [tex]\( \sqrt{d^{18}} = d^9 \)[/tex]
- Final expression: [tex]\( \frac{4 c^7}{9 d^9} \)[/tex].
5. Expression 23: [tex]\( \sqrt[3]{250 y^2 z^4} \)[/tex]
- Break down the cube root:
- [tex]\( 250 = 125 \cdot 2 = 5^3 \cdot 2 \)[/tex]
- [tex]\( \sqrt[3]{5^3 \cdot 2 \cdot y^2 \cdot z^4} = 5 \sqrt[3]{2 y^2 z^4} \)[/tex]
- Simplify [tex]\( z^4 \)[/tex]: [tex]\( \sqrt[3]{z^4} = z \cdot \sqrt[3]{z} \)[/tex]
- Final expression: [tex]\( 5 \cdot \sqrt[3]{2 y^2 z} \)[/tex].
These are the reduced radical forms for the first few given expressions. If you need the rest simplified, let me know, and I'll be happy to continue with them!
### Simplifying the Expressions:
1. Expression 19: [tex]\( (3 x^{\frac{1}{2}})(4 x^{\frac{3}{3}}) \)[/tex]
- Simplify the exponents: [tex]\( x^{\frac{3}{3}} = x^1 = x \)[/tex].
- Combine powers: [tex]\( x^{\frac{1}{2}} \cdot x^1 = x^{\frac{1}{2} + 1} = x^{\frac{3}{2}} \)[/tex].
- Multiply the coefficients: [tex]\( 3 \cdot 4 = 12 \)[/tex].
- Final expression: [tex]\( 12x^{\frac{3}{2}} \)[/tex].
2. Expression 20: [tex]\( 2 b^{\frac{1}{2}}(3 b^{\frac{1}{2}} c^{\frac{1}{3}})^2 \)[/tex]
- Expand the bracket using power: [tex]\( (3 b^{\frac{1}{2}} c^{\frac{1}{3}})^2 = 9 b^1 c^{\frac{2}{3}} \)[/tex].
- Multiply with [tex]\( 2 b^{\frac{1}{2}} \)[/tex]:
- Combine the powers of [tex]\( b \)[/tex]: [tex]\( b^{\frac{1}{2} + 1} = b^{\frac{3}{2}} \)[/tex].
- Coefficients multiply: [tex]\( 2 \cdot 9 = 18 \)[/tex].
- Final expression: [tex]\( 18b^{\frac{3}{2}}c^{\frac{2}{3}} \)[/tex].
3. Expression 21: [tex]\( \left(x^{\frac{1}{2}} \cdot x^{\frac{3}{12}}\right)^4 \div x^{\frac{2}{3}} \)[/tex]
- Simplify inside the power: [tex]\( x^{\frac{1}{2} + \frac{1}{4}} = x^{\frac{3}{4}} \)[/tex].
- Apply the exponent: [tex]\( (x^{\frac{3}{4}})^4 = x^3 \)[/tex].
- Divide by [tex]\( x^{\frac{2}{3}} \)[/tex]: [tex]\( x^{3 - \frac{2}{3}} = x^{\frac{7}{3}} \)[/tex].
- Final expression: [tex]\( x^{\frac{7}{3}} \)[/tex].
4. Expression 22: [tex]\( \left(\frac{16 c^{14}}{81 d^{18}}\right)^{\frac{1}{2}} \)[/tex]
- Take the square root:
- [tex]\( \sqrt{16} = 4 \)[/tex]
- [tex]\( \sqrt{81} = 9 \)[/tex]
- [tex]\( \sqrt{c^{14}} = c^7 \)[/tex]
- [tex]\( \sqrt{d^{18}} = d^9 \)[/tex]
- Final expression: [tex]\( \frac{4 c^7}{9 d^9} \)[/tex].
5. Expression 23: [tex]\( \sqrt[3]{250 y^2 z^4} \)[/tex]
- Break down the cube root:
- [tex]\( 250 = 125 \cdot 2 = 5^3 \cdot 2 \)[/tex]
- [tex]\( \sqrt[3]{5^3 \cdot 2 \cdot y^2 \cdot z^4} = 5 \sqrt[3]{2 y^2 z^4} \)[/tex]
- Simplify [tex]\( z^4 \)[/tex]: [tex]\( \sqrt[3]{z^4} = z \cdot \sqrt[3]{z} \)[/tex]
- Final expression: [tex]\( 5 \cdot \sqrt[3]{2 y^2 z} \)[/tex].
These are the reduced radical forms for the first few given expressions. If you need the rest simplified, let me know, and I'll be happy to continue with them!
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