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Chapter 6: Problem 60

Simplify the expression. Assume all variables are positive. \(\left(2 x \cdot 3 x^5\right)^3\)

Short Answer

Expert verified
The simplified expression is \(216x^{18}\).

Step by step solution

01

Perform multiplication inside brackets

The first multiplication is \(2x * 3x^5 = 6x^6\), by using the rule of exponents that states that when we multiply two powers of the same base, we add the exponents. Here, one \(x\) from \(2x\) (which is equal to \(x^1\)) is added to \(5\) from \(x^5\), to make a total of \(6\). So the expression becomes: \((6x^6)^3\).
02

Apply power of a power rule

Applying the power of a power rule, where we multiply the exponents, to the expression \((6x^6)^3\), we get: \(6^3 * (x^6)^3 = 216 * x^{18}\).
03

Group and state final answer

The final simplified expression is \(216x^{18}\). This is the most simplified form as there are no like terms to combine further.

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Key Concepts

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

Exponent Rules
To tackle algebraic expressions like \(\left(2 x \cdot 3 x^5\right)^3\), understanding exponent rules is crucial. Exponent rules guide us in simplifying expressions where numbers or variables are raised to powers. Let's break down these rules:
  • Product of Powers: If you multiply similar bases, you add their exponents. For instance, \(x^1 \times x^5\) simplifies to \(x^{1+5} = x^6\). This is because multiplying means you have the base repeated multiple times, and adding the exponents tells you the total number of times.
  • Power of a Power: When raising a power to another power, you multiply the exponents. For example, \((x^6)^3\) becomes \(x^{6 \times 3} = x^{18}\). This rule helps simplify expressions where powers themselves are raised to additional powers.
By using these rules, you can simplify even the most complex algebraic expressions more easily. Recognizing the right rule to apply at each step is the key to correct simplification.
Multiplication of Variables
Multiplying variables involves both their coefficients and their powers. In algebra, coefficients are the numerical parts while variables denote the letters like \(x\), \(y\), etc. In our expression, \(2x \cdot 3x^5\), consider both aspects:
  • Multiply Coefficients: Multiply the numerical coefficients. Using the example of \(2 \cdot 3\), the result is \(6\).
  • Combine Variable Terms: For variables like \(x\), add their exponents. The term \(x^1\) from \(2x\) and \(x^5\) from \(3x^5\) combine to form \(x^{1+5} = x^6\).
This process results in an expression like \(6x^6\), demonstrating both the combination of coefficients and variables. Ensuring you correctly multiply and add exponents prevents errors and leads to simpler expressions.
Power of a Power Rule
The power of a power rule is fundamental for simplifying expressions like \(6x^6)^3\). It explains how to manage situations where an already powered term is raised again.When you have an expression inside parentheses raised to an exponent, you apply the power to both the coefficient and the variable parts:
  • To the Coefficient: The numerical part like \(6\) in \((6x^6)^3\) is raised to the third power, resulting in \(6^3 = 216\).
  • To the Variable's Power: Multiply the existing exponent by the outer power. In our case, \(x^6\) raised to the third power becomes \(x^{6 \times 3} = x^{18}\).
Ultimately, the expression simplifies to \(216x^{18}\). Mastery of this rule allows for the seamless handling of multi-level exponents, keeping complex algebra manageable.

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Most popular questions from this chapter

In Exercises 27-30, use the properties of exponents to rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then find the percent rate of change. $$ y=0.5 e^{0.8 t} $$

When X-rays of a fixed wavelength strike a material \(x\) centimeters thick, the intensity \(I(x)\) of the X-rays transmitted through the material is given by \(I(x)=I_0 e^{-\mu x}\), where \(I_0\) is the initial intensity and \(\mu\) is a value that depends on the type of material and the wavelength of the X-rays. The table shows the values of \(\mu\) for various materials and X-rays of medium wavelength. $$ \begin{array}{|l|c|c|c|} \hline \text { Material } & \text { Aluminum } & \text { Copper } & \text { Lead } \\ \hline \text { Value of } \mu & 0.43 & 3.2 & 43 \\ \hline \end{array} $$a. Find the thickness of aluminum shielding that reduces the intensity of \(\mathrm{X}\)-rays to \(30 \%\) of their initial intensity. (Hint: Find the value of \(x\) for which \(I(x)=0.3 I_0\). b. Repeat part (a) for the copper shielding. c. Repeat part (a) for the lead shielding. d. Your dentist puts a lead apron on you before taking X-rays of your teeth to protect you from harmful radiation. Based on your results from parts (a)-(c), explain why lead is a better material to use than aluminum or copper.

Solve the equation. Check for extraneous solutions. \(\ln x+\ln (x-2)=5\)

You deposit \(\$ 100\) in an account that pays \(6 \%\) annual interest. How long will it take for the balance to reach \(\$ 1000\) for each frequency of compounding? a. annual b. quarterly c. daily d. continuously

Solve the equation. Check for extraneous solutions. \(\log _2 x+\log _2(x-2)=3\)
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