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Chapter 6: Problem 430

Mixed Practice $$\frac{27 a^{7}}{3 a^{3}}+\frac{54 a^{9}}{9 a^{5}}$$

Short Answer

Expert verified
15a^4

Step by step solution

01

- Simplify the first fraction

Consider the first fraction, \(\frac{27 a^{7}}{3 a^{3}}\). Simplify the constants and the variables separately. Divide 27 by 3 to get 9, and use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\): \[\frac{27 a^{7}}{3 a^{3}} = 9 a^{7-3} = 9a^4\]
02

- Simplify the second fraction

Consider the second fraction, \(\frac{54 a^{9}}{9 a^{5}}\). Again, simplify the constants and the variables separately. Divide 54 by 9 to get 6, and use the property of exponents again: \[\frac{54 a^{9}}{9 a^{5}} = 6 a^{9-5} = 6a^4\]
03

- Add the simplified terms

Add the simplified terms from Steps 1 and 2. Both terms have the same variable part \(a^4\), so they can be directly added together: \[9a^4 + 6a^4 = (9 + 6)a^4 = 15a^4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Fractions
Simplifying fractions in algebra can make complex expressions more manageable. To simplify a fraction, break it into smaller parts and reduce each part separately. For instance, consider the fraction \(\frac{27 a^{7}}{3 a^{3}}\). The steps are:
  • Simplify the constants: \( \frac{27}{3} = 9 \)
  • Apply the property of exponents: \(\frac{a^m}{a^n} = a^{m-n}\), so \( a^{7-3} = a^4 \)
Combining these, we get \( 9a^4 \). Repeat similar steps for the next fraction \(\frac{54 a^{9}}{9 a^{5}}\):
  • Simplify the constants: \( \frac{54}{9} = 6 \)
  • Apply the exponent rule: \( a^{9-5} = a^4 \)
This simplifies the fraction to \( 6a^4 \). Simplifying fractions helps reveal simpler, equivalent expressions.
Exponents Properties
Understanding exponents is crucial in algebra. Exponents indicate repeated multiplication, and certain properties make working with them easier. One such property is \( \frac{a^m}{a^n} = a^{m-n} \). This rule states that when you divide like bases, you subtract the exponents. For example, in the fraction \( \frac{27 a^{7}}{3 a^{3}} \), you can rewrite it by subtracting the exponent in the denominator from the exponent in the numerator: \( a^{7-3} = a^4 \). Similarly, for \( \frac{54 a^{9}}{9 a^{5}} \), you get \( a^{9-5} = a^4 \). This simplification is a result of the exponent properties, making the equation easier to handle.
Algebraic Addition
Algebraic addition combines like terms, which are terms with the same variable part. After simplifying fractions, like in \(\frac{27 a^{7}}{3 a^{3}} + \frac{54 a^{9}}{9 a^{5}}\), you get \( 9a^4 \) and \( 6a^4 \). Both have the same variable part \(a^4\). To add them:
  • Simply add their coefficients: \( 9 + 6 = 15 \)
Hence, \( 9a^4 + 6a^4 = 15a^4 \). Combining like terms simplifies the expression, giving \( 15a^4 \,\), which is easier to understand and use in further calculations.

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\frac{66 x^{3} y^{2}-110 x^{2} y^{3}-44 x^{4} y^{3}}{11 x^{2} y^{2}}$$

Mixed Practice $$\frac{56 y^{4} z^{5}}{7 y^{3} z^{3}}-\frac{45 y^{2} z^{2}}{5 y}$$

Mixed Practice (a) \(p^{4} \cdot p^{6}\) (b) \(\left(p^{4}\right)^{6}\)

Simplify. (a) \(3^{-4}\) (b) \(10^{-2}\)

Divide each polynomial by the binomial. $$\left(y^{2}+7 y+12\right) \div(y+3)$$
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