Question

using exponent rules simplify the problems (make sure your exponents are all positive)

1. \\(\frac{6m^{14}}{2m^7}\\)
2. \\((-4n^3)^0\\)
3. \\(6x^2 \cdot x^8 \cdot 5\\)
4. \\((3p^3)^2\\)
5. \\(\frac{12k^{14}}{24k^8}\\)

Explanation:

Problem 6: Simplify $\boldsymbol{\frac{6m^{14}}{2m^7}}$

Step1: Simplify the coefficients

Divide the coefficient of the numerator by the coefficient of the denominator: $\frac{6}{2} = 3$.

Step2: Apply the quotient rule for exponents

For the variable \( m \), use the rule \( \frac{a^m}{a^n}=a^{m - n} \). So, \( \frac{m^{14}}{m^7}=m^{14 - 7}=m^7 \).

Step3: Combine the results

Multiply the simplified coefficient and the simplified variable term: \( 3\times m^7 = 3m^7 \).

Step1: Apply the zero - exponent rule

Any non - zero number raised to the power of 0 is 1. The base \( - 4n^3 \) is non - zero (since if \( n = 0 \), the base is 0, but in general, for the zero - exponent rule, we consider non - zero bases. Assuming \( n
eq0 \) or if \( n = 0 \), the expression \( 0^0 \) is undefined, but for non - zero \( n \)), \( (-4n^3)^0=1 \).

Step1: Multiply the coefficients

Multiply the coefficients 6 and 5: \( 6\times5 = 30 \).

Step2: Apply the product rule for exponents

For the variable \( x \), use the rule \( a^m\cdot a^n=a^{m + n} \). So, \( x^2\cdot x^8=x^{2 + 8}=x^{10} \).

Step3: Combine the results

Multiply the simplified coefficient and the simplified variable term: \( 30\times x^{10}=30x^{10} \).

Answer:

\( 3m^7 \)

Problem 7: Simplify $\boldsymbol{(-4n^3)^0}$