Question

1. the expression $(x^2)(x^{-8})$ is equivalent to $x^n$. what is the value of $n$?
2. simplify the expression $(16k^8r^6)^{\frac{1}{2}}$.
3. simplify the expression $\frac{6.6 \times 10^6}{3.3 \times 10^{-3}}$.

Explanation:

Question 7

Step1: Recall exponent rule for multiplication

When multiplying two powers with the same base, we add the exponents: \(a^m \cdot a^n = a^{m + n}\). Here, the base is \(x\), \(m = 2\) and \(n=- 8\).

Step2: Apply the rule

\((x^{2})(x^{-8})=x^{2+( - 8)}\)
\(=x^{-6}\)
Since \((x^{2})(x^{-8}) = x^{n}\), then \(n=-6\)

Step1: Recall the power of a product rule and power of a power rule

The power of a product rule is \((ab)^n=a^n b^n\) and the power of a power rule is \((a^m)^n=a^{m\times n}\)

Step2: Apply power of a product rule

\((16k^{8}r^{6})^{\frac{1}{2}}=16^{\frac{1}{2}}(k^{8})^{\frac{1}{2}}(r^{6})^{\frac{1}{2}}\)

Step3: Simplify each term

We know that \(16^{\frac{1}{2}}=\sqrt{16} = 4\)
For \((k^{8})^{\frac{1}{2}}\), using power of a power rule: \(k^{8\times\frac{1}{2}}=k^{4}\)
For \((r^{6})^{\frac{1}{2}}\), using power of a power rule: \(r^{6\times\frac{1}{2}}=r^{3}\)

Step4: Combine the simplified terms

\(16^{\frac{1}{2}}(k^{8})^{\frac{1}{2}}(r^{6})^{\frac{1}{2}}=4k^{4}r^{3}\)

Step1: Recall the rule for dividing numbers in scientific notation

When dividing numbers in scientific notation \(\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}\)
Here, \(a = 6.6\), \(b = 3.3\), \(m = 6\) and \(n=-3\)

Step2: Divide the coefficients

\(\frac{6.6}{3.3}=2\)

Step3: Subtract the exponents of 10

\(10^{6-( - 3)}=10^{6 + 3}=10^{9}\)

Step4: Combine the results

\(\frac{6.6\times10^{6}}{3.3\times10^{-3}}=2\times10^{9}\)

Answer:

\(-6\)

Question 8