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Chapter 10: Problem 73

Simplify. $$ \left(8 x^{2} y^{8}\right)^{3} $$

Short Answer

Expert verified
The simplified form of the given expression is \(512 x^{6} y^{24}\)

Step by step solution

01

Step 1:Apply the exponent to each part inside the parentheses

Apply the exponent of 3 to each part inside the parentheses independently. It gives us: \(8^{3} (x^{2})^{3} (y^{8})^{3}\).
02

Simplify the exponential terms

Now, simplify each exponential term. As per the rules of exponents, when raising a power to a power, you multiply the exponents. \(8^{3}\) gives \(512\), \((x^{2})^{3}\) simplifies to \(x^{6}\), and \((y^{8})^{3}\) simplifies to \(y^{24}\).
03

Combine all the terms

After simplifying the terms, combine them all in one expression: \(512 x^{6} y^{24}\).

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Key Concepts

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

Exponential Rules
Understanding exponential rules is key to solving expressions involving exponents. Exponentiation is a mathematical operation that involves two numbers: the base and the exponent. The exponent indicates how many times the base is used as a factor in a multiplication. For example, in the expression \((x^a)^b\), the rule to recall is when raising a power to another power, you multiply the exponents, resulting in \(x^{a \times b}\). Learning these rules simplifies complex problems.

Some of the most important exponential rules include:
  • Product of Powers: When multiplying like bases, add the exponents: \(x^a \times x^b = x^{a+b}\).
  • Power of a Power: When raising a power to another power, multiply the exponents: \((x^a)^b = x^{ab}\).
  • Power of a Product: Distribute the exponent to each factor in the product: \((xy)^a = x^a y^a\).
Getting comfortable with these rules can make handling expressions and equations much simpler.
Simplifying Expressions
Simplifying expressions is the process of reducing them to their simplest form. It makes solving and understanding problems easier. In the exercise, we start with an expression like \((8x^2y^8)^3\). First, distribute the outer exponent of 3 to each component inside the parentheses.

Here鈥檚 how you simplify step-by-step:
  • Break Down: Apply the outer exponent to all terms: \(8^3(x^2)^3(y^8)^3\).
  • Calculate the Constant: Compute the outer exponent for the numerical base: \(8^3 = 512\).
  • Simplify with Exponents: Use the power of a power rule to simplify variables: \((x^2)^3 = x^6\) and \((y^8)^3 = y^{24}\).
Once simplified, combine all terms to get the clean expression \(512x^6y^{24}\). This method is vital for managing more complex expressions with multiple variables.
Multiplying Exponents
Multiplying exponents involves specific rules that make it straightforward. When you multiply terms that have the same base, you simply add the exponents. This rule keeps things neat and manageable. In our example, while simplifying \((x^2)^3\), use the power of a power rule, which involves multiplying the exponents

Here's how it works:
  • Identify the Bases: Look for common bases in the expression.
  • Apply Multiplication Rule: Use \((x^a)^b = x^{ab}\) to multiply the exponents, where \( a \) and \( b \) are both powers multiplied together.
  • Final Combine: When your terms are simplified, you express them with a single exponent: \((x^2)^3 = x^{6}\).
This approach is essential for efficiently handling powers in larger algebraic expressions. Remember, practicing these exponent rules will make any future calculations much smoother.

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Most popular questions from this chapter

An object is propelled from the ground with an initial upward velocity of 224 feet per second. Will the object reach a height of 784 feet? If it does, how long will it take the object to reach that height? Solve by factoring.

Use factoring to solve the equation. Use a graphing calculator to check your solution if you wish. $$90 x^{2}-120 x+40=0$$

Which one of the following is a correct factorization of the expression \(-16 x^{2}+36 x+52 ?\). A) \((-16 x+26)(x-2)\) (B) \((-4 x-13)(4 x+4)\) (C) \(-1(4 x-13)(4 x+4)\) (D) \(-1(4 x-4)(4 x+13)\)

Use the following information. In the sport of pole-vaulting, the height \(h\) (in feet) reached by a pole- vaulter is a function of \(v,\) the velocity of the pole-vaulter, as shown in the model below. The constant \(g\) is approximately 32 feet per second per second. Pole-vaulter height model: \(h=\frac{v^{2}}{2 g}\) To reach a height of 9 feet, what is the pole-vaulter's velocity?

Use a is calculator to evaluate the expression. Round the result to two decimal places when appropriate. $$\left(-6 \cdot 3^{4}\right)^{3}$$
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