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Chapter 0: Problem 95

Simplify using properties of exponents. $$\left(x^{\frac{2}{3}}\right)^{3}$$

Short Answer

Expert verified
The simplified form of the given expression is \(x^{2}\).

Step by step solution

01

Apply the power of a power property

Using the power of a power property, the expression can be rewritten as: \(x^{(\frac{2}{3}*3)}\). This simplifies the expression by getting rid of the brackets.
02

Simplify the expression

In the step 1, the expression was rewritten as \(x^{(\frac{2}{3}*3)}\). Now simplify the exponent: \((\frac{2}{3}*3) = 2\). So, the expression is simplified to: \(x^{2}\).

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Key Concepts

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

Power of a Power Property
The "power of a power" property is a fundamental concept in exponents. It states that when you raise an exponential expression to another power, you multiply the exponents. This is written as:
  • \((a^m)^n = a^{m \cdot n}\)
For instance, in the expression \((x^{\frac{2}{3}})^3\), you apply this property by multiplying the outer exponent (3) with the inner exponent (\(\frac{2}{3}\)). This results in a simplified exponent of \((\frac{2}{3} \times 3)\). Breaking it down:
  • The inner exponent: \(\frac{2}{3}\)
  • The outer exponent: 3
  • Multiply them: \(\frac{2}{3} \times 3 = 2\)
Remember, this property helps streamline expressions by eliminating brackets and reducing complexity. The resulting exponent is much easier to work with!
Simplification
Simplification is all about making an expression easier to work with or understand without changing its value. In our exercise, the goal is to remove the brackets and deal with the exponents.
  • Start with: \((x^{\frac{2}{3}})^3\)
  • Apply the power of a power property: \(x^{(\frac{2}{3} \cdot 3)}\)
  • Simplify the multiplication: \(x^2\)
This process of simplification is essential in algebra. It transforms complicated expressions into something neater and more manageable. Practicing these steps with various problems can deepen your understanding and efficiency in solving exponential expressions.
Exponential Expressions
Exponential expressions are mathematical expressions that involve exponents. They represent repeated multiplication of a base number. For example, \(x^3\) means we multiply \(x\) by itself three times.In solving problems involving exponential expressions, some key points to remember are:
  • Exponents only apply to the number directly before them, unless parentheses are involved.
  • Use properties like the "power of a power" to simplify.
  • Learn to identify bases and exponents quickly.
In the expression \((x^{\frac{2}{3}})^3\), the base is \(x\), and the process of simplifying helps us see that this can be reduced to \(x^2\). This type of understanding is crucial when dealing with more complex math scenarios.

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Most popular questions from this chapter

If you are given a quadratic equation, how do you determine which method to use to solve it?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although I can solve \(3 x+\frac{1}{5}=\frac{1}{4}\) by first subtracting \(\frac{1}{5}\) from both sides, I find it easier to begin by multiplying both sides by \(20,\) the least common denominator.

Factor completely. $$x^{4}-y^{4}-2 x^{3} y+2 x y^{3}$$

How is the quadratic formula derived?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\)
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