Problem 6 - Algebra and Trigonometry

Question

Find the missing power in the following calculation: \(5^{1 / 3} \cdot 5=5 .\)

Step-by-Step Solution

Answer

The setup implies a simplification error; redo and check calculations.

Step 1: Identify the Base

Notice that both sides of the equation involve the number 5, indicating that 5 is the base of the power.

Step 2: Understanding Power Properties

Remember that when multiplying two powers with the same base, you add the exponents. Therefore, \[ a^m imes a^n = a^{m+n} \]In our problem, this translates to:\[ 5^{1/3} imes 5^1 = 5^{1/3 + 1} \]

Step 3: Simplifying the Right Side

The right side of the equation is 5, which is equivalent to:\[ 5^1 \]This means that\[ 5^{1/3 + 1} = 5^1 \]

Step 4: Setting Exponents Equal

Since the bases are equal, the exponents must be equal. Therefore, we have:\[ 1/3 + 1 = 1 \]

Step 5: Solve for the Missing Exponent

Solve the equation by subtracting 1 on both sides:\[ \frac{1}{3} + 1 = 1 \]\[ \frac{1}{3} = 1 - 1 \]\[ \frac{1}{3} = 0 \]

Step 6: Conclusion: Error in Initial Setup

After solving, you realize it doesn't make sense for \( \frac{1}{3} \) to equal 0. Re-evaluate the problem: if originally it was set up correctly, review any misjudgment. Since \( 1 eq \frac{1}{3} + 1 \), reconsider missing powers on either side.

Key Concepts

Power Properties

Understanding the properties of powers or exponents is vital to simplifying and solving expressions. A fundamental property is that when you multiply powers with the same base, you add the exponents. This can be written as:

• \( a^m \times a^n = a^{m+n} \)

This property helps simplify expressions and solve equations where the bases are the same.
When you encounter multiplication involving exponents, always look out for the shared base, as this is your ticket to combining those powers efficiently.
In our original problem, because both numbers are powers of 5, we can apply this property directly. This makes the process of solving the equation much more manageable.

Simplifying Exponents

Simplifying exponents is essential for making math expressions more understandable. Sometimes when facing a complicated expression, breaking it down can make things easier.
Take a look at the original equation:

• \( 5^{1/3} \cdot 5 = 5 \)

Notice that \( 5 \) can be written as \( 5^1 \). This is an important step in simplifying because it allows us to express all parts of the equation consistently.
Once the bases are the same, we use power properties to add those exponents. This process is critical in solving the equation and finding the missing part.

Base Identification

Identifying the correct base in an exponent expression is like finding the common thread that connects all parts of the equation.
In the exercise given, the base is the number that remains the same while the exponents change or need to be solved.
For instance, in \( 5^{1/3} \cdot 5 = 5 \), the base is clearly 5.

• We identify this because the problem features 5 consistently.
• This identification helps in using power properties effectively.

Once you have spotted the base, you can manipulate the exponents easily, which is a crucial step in simplifying and solving exponent-related problems.

Exponent Addition

Exponent addition is a straightforward process once you've got the hang of the basic power property. Whenever you multiply numbers with the same base, you simply add the exponents together.
In our example:

• You're multiplying \( 5^{1/3} \) and \( 5 \) (which is \( 5^1 \)).
• Thus, \( 1/3 + 1 \) becomes \( 1/3 + 3/3 = 4/3 \)

This allows you to create a single, more simplified exponent.
Remember, exponent addition only works when the bases are the same; otherwise, you'll need a different approach to tackle the equation.