Question

1. simplify: \\(\frac{24x^{5}y^{8}}{3x^{-3}y^{2}}\\).

1. write an expression equivalent to \\((3a^{0}b^{4}c^{20})(4a^{5}b^{-15}c^{0})\\) for all values of a, b, and c where the expression is defined.

1. simplify the expression \\((rs^{-3})^{4}\\).

1. the expression \\((y^{12})(z^{3})^{2}\\) is equivalent to \\(y^{p}\\). what is the value of p?

1. the radius of a circle is \\(9m^{6}n^{4}\\) inches. what is the area of the circle? use \\(a = \pi r^{2}\\), leave in terms of \\(\pi\\).

1. simplify the expression \\((8x^{9})^{2}(x^{6})^{\frac{1}{2}}\\).

Explanation:

Problem 1

Step1: Divide the coefficients

Divide 24 by 3: $\frac{24}{3} = 8$

Step2: Simplify the \(x\)-terms

Using the rule \( \frac{x^m}{x^n}=x^{m - n} \), for \(x\)-terms: \(x^{5-(-3)} = x^{5 + 3}=x^8\)

Step3: Simplify the \(y\)-terms

Using the rule \( \frac{y^m}{y^n}=y^{m - n} \), for \(y\)-terms: \(y^{8-2}=y^6\)

Step4: Combine the results

Multiply the coefficient, \(x\)-term, and \(y\)-term together: \(8\times x^8\times y^6 = 8x^8y^6\)

Step1: Multiply the coefficients

Multiply 3 and 4: \(3\times4 = 12\)

Step2: Simplify the \(a\)-terms

Using the rule \(a^m\times a^n=a^{m + n}\), for \(a\)-terms: \(a^{0 + 5}=a^5\)

Step3: Simplify the \(b\)-terms

Using the rule \(b^m\times b^n=b^{m + n}\), for \(b\)-terms: \(b^{4+(-15)}=b^{-11}\)

Step4: Simplify the \(c\)-terms

Using the rule \(c^m\times c^n=c^{m + n}\), for \(c\)-terms: \(c^{20+0}=c^{20}\)

Step5: Combine the results

Multiply the coefficient, \(a\)-term, \(b\)-term, and \(c\)-term together: \(12\times a^5\times b^{-11}\times c^{20}=\frac{12a^5c^{20}}{b^{11}}\) (since \(b^{-11}=\frac{1}{b^{11}}\))

Step1: Apply the power of a product rule

Using \((ab)^n=a^n\times b^n\), we get \(r^4\times(s^{-3})^4\)

Step2: Apply the power of a power rule

Using \((a^m)^n=a^{m\times n}\), for \((s^{-3})^4\): \(s^{-3\times4}=s^{-12}\)

Step3: Combine the results

We have \(r^4\times s^{-12}=\frac{r^4}{s^{12}}\) (since \(s^{-12}=\frac{1}{s^{12}}\))

Answer:

\(8x^8y^6\)

Problem 2