Question

1. simplify: \\(\frac{14a^{4}b^{5}c^{-10}}{8a^{-3}b^{4}c^{-5}}\\)\
2. the area of a rectangle is \\(54x^{9}y^{8}\\) square yards. if the length of the rectangle is \\(6x^{3}y^{4}\\) yards, what is the width of the rectangle in yards?\
3. a circle has a radius of \\(6x^{9}y^{5}\\) cm. the area of a circle can be found using \\(a = \pi r^{2}\\). what is the area of this circle in square centimeters?\
4. simplify: \\((7x^{\frac{\pi}{2}})^{2}(x^{8})^{\frac{\pi}{2}}\\)

Explanation:

Problem 1: Simplify \(\frac{14a^{4}b^{5}c^{-10}}{8a^{-3}b^{4}c^{-5}}\)

Step 1: Simplify the coefficients

Simplify the fraction of the coefficients \(14\) and \(8\) by dividing both by their greatest common divisor, which is \(2\).
\(\frac{14}{8}=\frac{7}{4}\)

Step 2: Simplify the \(a\)-terms using exponent rules

For the \(a\)-terms, use the rule \( \frac{a^{m}}{a^{n}}=a^{m - n}\). Here, \(m = 4\) and \(n=- 3\), so \(a^{4-(-3)}=a^{4 + 3}=a^{7}\)

Step 3: Simplify the \(b\)-terms using exponent rules

For the \(b\)-terms, use the rule \( \frac{b^{m}}{b^{n}}=b^{m - n}\). Here, \(m = 5\) and \(n = 4\), so \(b^{5-4}=b^{1}=b\)

Step 4: Simplify the \(c\)-terms using exponent rules

For the \(c\)-terms, use the rule \( \frac{c^{m}}{c^{n}}=c^{m - n}\). Here, \(m=-10\) and \(n = - 5\), so \(c^{-10-(-5)}=c^{-10 + 5}=c^{-5}=\frac{1}{c^{5}}\)

Step 5: Combine all the simplified terms

Multiply the simplified coefficient, \(a\)-term, \(b\)-term, and \(c\)-term together: \(\frac{7}{4}\times a^{7}\times b\times\frac{1}{c^{5}}=\frac{7a^{7}b}{4c^{5}}\)

Step 1: Recall the formula for the area of a rectangle

The area of a rectangle \(A=l\times w\), where \(l\) is the length and \(w\) is the width. So, \(w=\frac{A}{l}\)

Step 2: Substitute the given values into the formula

We know that \(A = 54x^{9}y^{8}\) and \(l=6x^{3}y^{4}\). So, \(w=\frac{54x^{9}y^{8}}{6x^{3}y^{4}}\)

Step 3: Simplify the coefficient

Simplify the fraction of the coefficients \(54\) and \(6\): \(\frac{54}{6} = 9\)

Step 4: Simplify the \(x\)-terms using exponent rules

For the \(x\)-terms, use the rule \( \frac{x^{m}}{x^{n}}=x^{m - n}\). Here, \(m = 9\) and \(n = 3\), so \(x^{9-3}=x^{6}\)

Step 5: Simplify the \(y\)-terms using exponent rules

For the \(y\)-terms, use the rule \( \frac{y^{m}}{y^{n}}=y^{m - n}\). Here, \(m = 8\) and \(n = 4\), so \(y^{8-4}=y^{4}\)

Step 6: Combine the simplified terms

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term together: \(9\times x^{6}\times y^{4}=9x^{6}y^{4}\)

Step 1: Recall the formula for the area of a circle

The area of a circle \(A=\pi r^{2}\), where \(r\) is the radius of the circle.

Step 2: Substitute the given radius into the formula

We know that \(r = 6x^{9}y^{5}\). So, \(A=\pi(6x^{9}y^{5})^{2}\)

Step 3: Apply the exponent rule \((ab)^{n}=a^{n}b^{n}\)

\((6x^{9}y^{5})^{2}=6^{2}\times(x^{9})^{2}\times(y^{5})^{2}\)

Step 4: Simplify each term

• \(6^{2}=36\)
• Using the rule \((a^{m})^{n}=a^{mn}\), \((x^{9})^{2}=x^{9\times2}=x^{18}\)
• Using the rule \((a^{m})^{n}=a^{mn}\), \((y^{5})^{2}=y^{5\times2}=y^{10}\)

Step 5: Combine the terms

Multiply the simplified terms together: \(A=\pi\times36\times x^{18}\times y^{10}=36\pi x^{18}y^{10}\)

Answer:

\(\frac{7a^{7}b}{4c^{5}}\)

Problem 2: Find the width of the rectangle