Explore a diverse range of topics and get expert answers on IDNLearn.com. Ask anything and receive prompt, well-informed answers from our community of experienced experts.
Sagot :
To solve the problem, we need to simplify the expression [tex]\(\frac{2 x^{-\frac{1}{4}} y^2}{\sqrt[4]{81 x^5 y^2}}\)[/tex] and rewrite it in the form [tex]\(a x^b y^c\)[/tex]. We will then find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] and calculate their product.
1. Rewrite the expression:
[tex]\[ \frac{2 x^{-\frac{1}{4}} y^2}{\sqrt[4]{81 x^5 y^2}} \][/tex]
2. Simplify the denominator using properties of exponents and roots:
[tex]\[ \sqrt[4]{81 x^5 y^2} = (81 x^5 y^2)^{\frac{1}{4}} \][/tex]
Since [tex]\(81 = 3^4\)[/tex], we have:
[tex]\[ (81 x^5 y^2)^{\frac{1}{4}} = 3 (x^5)^{\frac{1}{4}} (y^2)^{\frac{1}{4}} = 3 x^{\frac{5}{4}} y^{\frac{1}{2}} \][/tex]
3. Simplify the entire expression by dividing the numerator by the simplified denominator:
[tex]\[ \frac{2 x^{-\frac{1}{4}} y^2}{3 x^{\frac{5}{4}} y^{\frac{1}{2}}} \][/tex]
4. Combine the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ = \frac{2}{3} \cdot \frac{x^{-\frac{1}{4}}}{x^{\frac{5}{4}}} \cdot \frac{y^2}{y^{\frac{1}{2}}} \][/tex]
For the [tex]\(x\)[/tex] terms:
[tex]\[ x^{-\frac{1}{4}} \cdot x^{-\frac{5}{4}} = x^{-\frac{1}{4} - \frac{5}{4}} = x^{-\frac{6}{4}} = x^{-\frac{3}{2}} \][/tex]
For the [tex]\(y\)[/tex] terms:
[tex]\[ y^2 \cdot y^{-\frac{1}{2}} = y^{2 - \frac{1}{2}} = y^{\frac{3}{2}} \][/tex]
5. Combine all parts:
[tex]\[ \frac{2}{3} \cdot x^{-\frac{3}{2}} \cdot y^{\frac{3}{2}} \][/tex]
Therefore, the expression has been simplified to:
[tex]\[ \frac{2}{3} \cdot x^{-\frac{3}{2}} \cdot y^{\frac{3}{2}} \][/tex]
So, we have [tex]\(a = \frac{2}{3}\)[/tex], [tex]\(b = -\frac{3}{2}\)[/tex], and [tex]\(c = \frac{3}{2}\)[/tex].
6. Calculate the product [tex]\(a \cdot b \cdot c\)[/tex]:
[tex]\[ \left(\frac{2}{3}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(\frac{3}{2}\right) \][/tex]
Multiplying these together:
[tex]\[ \left(\frac{2}{3} \cdot \frac{3}{2}\right) = 1 \quad \text{and} \quad 1 \cdot \left(\frac{3}{2}\right) = \left(\frac{3}{2}\right) \][/tex]
[tex]\[ \left(\frac{3}{2}\right) \cdot \left(-1\right) = 0 \][/tex]
Thus, the product of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is:
[tex]\[ 0 \][/tex]
Therefore, the product is [tex]\(\boxed{0}\)[/tex].
1. Rewrite the expression:
[tex]\[ \frac{2 x^{-\frac{1}{4}} y^2}{\sqrt[4]{81 x^5 y^2}} \][/tex]
2. Simplify the denominator using properties of exponents and roots:
[tex]\[ \sqrt[4]{81 x^5 y^2} = (81 x^5 y^2)^{\frac{1}{4}} \][/tex]
Since [tex]\(81 = 3^4\)[/tex], we have:
[tex]\[ (81 x^5 y^2)^{\frac{1}{4}} = 3 (x^5)^{\frac{1}{4}} (y^2)^{\frac{1}{4}} = 3 x^{\frac{5}{4}} y^{\frac{1}{2}} \][/tex]
3. Simplify the entire expression by dividing the numerator by the simplified denominator:
[tex]\[ \frac{2 x^{-\frac{1}{4}} y^2}{3 x^{\frac{5}{4}} y^{\frac{1}{2}}} \][/tex]
4. Combine the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ = \frac{2}{3} \cdot \frac{x^{-\frac{1}{4}}}{x^{\frac{5}{4}}} \cdot \frac{y^2}{y^{\frac{1}{2}}} \][/tex]
For the [tex]\(x\)[/tex] terms:
[tex]\[ x^{-\frac{1}{4}} \cdot x^{-\frac{5}{4}} = x^{-\frac{1}{4} - \frac{5}{4}} = x^{-\frac{6}{4}} = x^{-\frac{3}{2}} \][/tex]
For the [tex]\(y\)[/tex] terms:
[tex]\[ y^2 \cdot y^{-\frac{1}{2}} = y^{2 - \frac{1}{2}} = y^{\frac{3}{2}} \][/tex]
5. Combine all parts:
[tex]\[ \frac{2}{3} \cdot x^{-\frac{3}{2}} \cdot y^{\frac{3}{2}} \][/tex]
Therefore, the expression has been simplified to:
[tex]\[ \frac{2}{3} \cdot x^{-\frac{3}{2}} \cdot y^{\frac{3}{2}} \][/tex]
So, we have [tex]\(a = \frac{2}{3}\)[/tex], [tex]\(b = -\frac{3}{2}\)[/tex], and [tex]\(c = \frac{3}{2}\)[/tex].
6. Calculate the product [tex]\(a \cdot b \cdot c\)[/tex]:
[tex]\[ \left(\frac{2}{3}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(\frac{3}{2}\right) \][/tex]
Multiplying these together:
[tex]\[ \left(\frac{2}{3} \cdot \frac{3}{2}\right) = 1 \quad \text{and} \quad 1 \cdot \left(\frac{3}{2}\right) = \left(\frac{3}{2}\right) \][/tex]
[tex]\[ \left(\frac{3}{2}\right) \cdot \left(-1\right) = 0 \][/tex]
Thus, the product of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is:
[tex]\[ 0 \][/tex]
Therefore, the product is [tex]\(\boxed{0}\)[/tex].
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.
