Problem 46 - Precalculus

Question

Show that \(\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}}=x\) for all \(x \geq 0\).

Step-by-Step Solution

Answer

The expression simplifies to \(x\) for all \(x \geq 0\).

Step 1: Understanding the Given Expression

The given expression is \((x^{\frac{3}{2}})^{\frac{2}{3}}\). This expression implies that we first raise \(x\) to the power of \(\frac{3}{2}\) and then raise the result to the power \(\frac{2}{3}\).

Step 2: Apply the Power Rule of Exponents

According to the power rule, \((a^m)^n = a^{m \cdot n}\). We apply this rule to the expression \((x^{\frac{3}{2}})^{\frac{2}{3}}\), which gives us the equality: \[ (x^{\frac{3}{2}})^{\frac{2}{3}} = x^{\frac{3}{2} \cdot \frac{2}{3}} \] This multiplication of exponents simplifies the expression.

Step 3: Simplify the Power Expression

Now, multiply the exponents: \[ \frac{3}{2} \cdot \frac{2}{3} = \frac{3 \times 2}{2 \times 3} = \frac{6}{6} = 1 \] Substitute this result back into the power expression: \[ x^{\frac{6}{6}} = x^1 = x \]

Step 4: Conclusion

We have shown that \((x^{\frac{3}{2}})^{\frac{2}{3}} = x\) by applying exponent rules and simplifying the expression. This holds for all \(x \geq 0\) as the original bases and operations are defined for non-negative real numbers.

Key Concepts

Understanding the Power Rule

The power rule is a helpful tool when dealing with expressions that include exponents raised to further exponents. According to this rule, when you have a power raised to another power, you simply multiply the exponents. For example,

• If you have \[ (a^m)^n \], you can rewrite this as \[ a^{m imes n} \].
• This simplifies the expression by reducing the nested exponents into a single one.

It's crucial to note that the power rule only applies when the bases are the same. This fundamental technique makes it easier to handle complex exponentiation problems.
In our original problem, the expression \[ (x^{\frac{3}{2}})^{\frac{2}{3}} \] uses the power rule to become \[ x^{\frac{3}{2} \times \frac{2}{3}} \]. This simplification reduces the problem to a straightforward exponent multiplication task.

Clear Explanation of Exponents

Exponents, also known as powers, indicate how many times a number, known as the base, is multiplied by itself. An easy way to visualize this is:

• For positive whole number exponents: \[ a^n \]neans multiplying the base \( a \) by itself \( n \) times.
• For fractions as exponents: \[ a^{m/n} \], it implies you take the \( n \)-th root of the base raised to the \( m \)-th power.

This helps manage operations on large numbers easily or for simplifying expressions that appear complicated.
In the expression \( x^{\frac{3}{2}} \), \( 3/2 \) provides a way to express a square root and cubing in a single operation, adding a layer of flexibility and power to mathematical calculations.

Techniques for Simplifying Expressions

Simplifying expressions involves reducing them to their simplest or most manageable form. This can include eliminating unnecessary components, combining like terms, or making use of mathematical properties and rules to rewrite expressions more compactly.

• When working with exponents, simplify by using the power rule or other exponent rules, such as:\[ a^m \times a^n = a^{m+n} \] or \[ \frac{a^m}{a^n} = a^{m-n} \].
• Another key is recognizing perfect powers or roots to reduce calculations.

In our exercise, applying the right rules and calculating the product of the exponents \( \frac{3}{2} \times \frac{2}{3} \) resulted in \( x^1 \) or simply \( x \). This demonstrates the power and simplicity in handling sophisticated operations through systematic steps.