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Chapter 1: Problem 37

Simplify the expression and climinate any negative exponent(s). $$\frac{\left(6 y^{3}\right)^{4}}{2 y^{5}}$$

Short Answer

Expert verified
The simplified expression is \(648y^7\).

Step by step solution

01

Apply the Power Rule

First, apply the power rule \((a^m)^n = a^{m\cdot n}\)to simplify the expression in the numerator. This results in:\((6y^3)^4 = 6^4\cdot (y^3)^4 = 1296y^{12}\).
02

Rewrite the Fraction

Rewrite the fraction with the new expression:\[\frac{1296y^{12}}{2y^5}\].
03

Simplify the Coefficient

Divide the coefficients in the fraction:\[\frac{1296}{2} = 648\].
04

Apply the Quotient Rule for Exponents

Use the quotient rule for exponents \(\frac{a^m}{a^n} = a^{m-n}\)to simplify the variable's exponents:\[\frac{y^{12}}{y^5} = y^{12-5} = y^7\].
05

Combine the Results

Combine the simplified coefficient and variable expression to get the final simplified form:\[648y^7\].

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Key Concepts

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

Power Rule
The power rule is a fundamental concept when it comes to simplifying expressions that involve exponents. It states that when you have an expression in the form of \(\( (a^m)^n = a^{m \cdot n} \)\), you multiply the exponents together. This is crucial when you're dealing with expressions raised to another power, as it allows you to simplify them.
Imagine the original expression \(\((6y^3)^4\)\). Here, both the constant 6 and the variable \(y\) with its exponent are raised to the 4th power. We apply the power rule:
  • Calculate \(6^4\), which is \(1296\).
  • For \(y^3\) raised to 4, multiply the exponents: \(3 \times 4 = 12\).
This simplifies the expression inside the parenthesis to \(1296y^{12}\), a necessary step before moving onto the next simplification process.
Quotient Rule
The quotient rule helps us deal with expressions where the numerator and denominator have the same base. It's stated as \(\(\frac{a^m}{a^n} = a^{m-n}\)\). By using this rule, we subtract the exponent of the denominator from the exponent of the numerator when they have the same base.
In our simplified expression, \(\(\frac{1296y^{12}}{2y^5}\)\), after separating the coefficients, we focus on the variables:
  • We treat \(y^{12}\) as the numerator and \(y^{5}\) as the denominator.
  • Apply the quotient rule by subtracting the exponents: \(12 - 5 = 7\).
Thus, the expression simplifies to \(y^7\), resulting in a much simpler format. This rule ensures we efficiently handle expressions involving division of the same base with exponents.
Negative Exponents
Negative exponents may at first seem confusing, but understanding them is made simple by grasping their basic principle. A negative exponent indicates that the base should be taken as the reciprocal. In formula terms, \(a^{-n} = \frac{1}{a^n}\).
While in this problem, explicit negative exponents were not directly presented, knowing how to eliminate them is a vital skill,especially useful when a negative exponent appears in a denominator or numerator during simplifications.
  • Understand that \(x^{-n}\) shifts to the opposite part of the fraction.
  • Place it as its positive counterpart, moving from numerator to denominator, or vice versa.
This knowledge is key when encountering negative exponents, giving you the tools to simplify and convert them into positive exponents efficiently, ensuring clear, simplified final expressions.

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Most popular questions from this chapter

Use the Difference of Squares Formula to factor \(17^{2}-16^{2}\). Notice that it is easy to calculate the factored form in your head, but not so easy to calculate the original form in this way. Evaluate each expression in your head: (a) \(528^{2}-527^{2}\) (b) \(122^{2}-120^{2}\) (c) \(1020^{2}-1010^{2}\) Now use the Special Product Formula $$(A+B)(A-B)=A^{2}-B^{2}$$ to evaluate these products in your head: (d) \(79 \cdot 51\) (e) \(998 \cdot 1002\)

Factor the expression completely. $$27 a^{3}-b^{6}$$

Factor the expression completely. $$\left(a^{2}+2 a\right)^{2}-2\left(a^{2}+2 a\right)-3$$

Factor the expression completely. $$r^{2}-6 r s+9 s^{2}$$

Factor the expression completely. $$9 x^{2}-36 x-45$$
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