Chapter 5: Problem 20

Simplify each of the following: (a) \(\frac{m^{-7}}{m^{-4}}\) (b) \(\left(3 a b^{2} c\right)^{3}\) (c) \(\left.y^{3} \times y^{-2} \times x^{7} \times x^{5} \times x^{-3}\right)\)

Short Answer

Expert verified

b) What is the simplified form of \(\left(3 a b^{2} c\right)^{3}\)? c) What is the simplified form of \(y^{3} \times y^{-2} \times x^{7} \times x^{5} \times x^{-3}\)?

Step by step solution

(a) Simplify \(\frac{m^{-7}}{m^{-4}}\)

We can simplify this expression by using the exponent division rule, which is \(\frac{a^m}{a^n} = a^{m-n}\). For our case, the rule becomes \(\frac{m^{-7}}{m^{-4}} = m^{-7 - (-4)}\). Applying the rule and combining the exponents, we get: \(m^{-7 - (-4)} = m^{-7+4} = m^{-3}\). So, the simplified form is \(m^{-3}\).

(b) Simplify \(\left(3 a b^{2} c\right)^{3}\)

To simplify this expression, we can use the power of a power exponent rule, which is \((a^m)^n = a^{m*n}\). We apply this rule to each term inside the parentheses: \(\left(3 a b^{2} c\right)^{3} = 3^3 \times a^3 \times (b^2)^3 \times c^3\). Now, we calculate \(3^3\) and apply the power of a power rule to \((b^2)^3\): \(3^3 \times a^3 \times (b^2)^3 \times c^3 = 27a^3\times b^{2*3} \times c^3\). Finally, we simplify and obtain: \(27a^3\times b^{6} \times c^3\). So, the simplified form is \(27a^3b^6c^3\).

(c) Simplify \(y^{3} \times y^{-2} \times x^{7} \times x^{5} \times x^{-3}\)

We can simplify this expression by combining the exponents of the same base using the multiplication exponent rule, which is \(a^m \cdot a^n = a^{m+n}\). We start by combining the exponents for base \(y\): \(y^{3} \cdot y^{-2} = y^{3-2} = y^1\). Next, we combine the exponents for base \(x\): \(x^{7} \cdot x^{5} \cdot x^{-3} = x^{7+5-3} = x^9\). Finally, we combine the simplified terms: \(y^1 \cdot x^9 = yx^9\). So, the simplified form is \(yx^9\).

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Simplify each of the following: (a) \(\frac{m^{-7}}{m^{-4}}\) (b) \(\left(3 a b^{2} c\right)^{3}\) (c) \(\left.y^{3} \times y^{-2} \times x^{7} \times x^{5} \times x^{-3}\right)\)

Short Answer

Step by step solution

(a) Simplify \(\frac{m^{-7}}{m^{-4}}\)

(b) Simplify \(\left(3 a b^{2} c\right)^{3}\)

(c) Simplify \(y^{3} \times y^{-2} \times x^{7} \times x^{5} \times x^{-3}\)

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