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Chapter 5: Problem 106

Simplify. $$ y^{4 x} \cdot y^{2 x} $$

Short Answer

Expert verified
y^{6x}

Step by step solution

01

- Identify the base and exponents

Observe that both terms have the same base, which is y. The exponents are 4x and 2x respectively.
02

- Apply the product of powers rule

Use the product of powers rule, which states that when multiplying two exponents with the same base, you add the exponents together: \[ y^{a} \times y^{b} = y^{a+b} \]
03

- Add the exponents

Add the exponents 4x and 2x together: \[ y^{4x + 2x} \]
04

- Simplify the expression

Combine the exponents: \[ y^{6x} \]

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Key Concepts

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

Product of Powers Rule Explained
Simplifying expressions with exponents can be easy if you know the right rules. One of the key rules is the Product of Powers rule.
When you multiply two terms that have the same base, you can simplify them by adding their exponents together.
The formula looks like this: \( y^{a} \times y^{b} = y^{a+b} \).
This rule takes advantage of the fact that you are essentially combining all the instances of the base number by merging their exponents.
  • For example, in the expression \( y^{4x} \times y^{2x} \), the base is 'y'.
  • The exponents are 4x and 2x.
Using the product of powers rule, we can simplify this expression by adding the exponents: \( y^{4x+2x} \), which then becomes \( y^{6x} \).
Adding Exponents
Adding exponents is straightforward but vital for simplifying expressions.
When you have the same base in your expression, you can focus on the exponents.
In our example, the exponents are 4x and 2x.
By the product of powers rule, you add these exponents together.
  • Let's look at the math step-by-step: you have two exponents, 4x and 2x.
  • Adding them together gives you: 4x + 2x = 6x.
So, \( y^{4x} \times y^{2x} \) simplifies to \( y^{4x + 2x} \). Finally, this becomes \( y^{6x} \).
This simple addition helps transform the expression into a more manageable form.
Same Base Multiplication
Same base multiplication is an important concept that makes understanding exponents easier.
If you are multiplying terms with the same base, you only need to add their exponents.
In the example \( y^{4x} \times y^{2x} \), the base is 'y', which is the same for both terms.
  • Since the bases are the same, you only need to focus on the exponents: 4x and 2x.
  • By adding these exponents (4x + 2x), you get 6x.
This means: \( y^{4x} \times y^{2x} \) simplifies to \( y^{6x} \).
Therefore, always remember that with same base multiplication, adding the exponents is your key to simplification.

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