91Ó°ÊÓ

When working with exponents, the Power of a Power Rule is a must-know. This rule helps simplify expressions where an exponent is raised to another exponent.
To put it plainly, it states that

• when you have something like \[(a^m)^n\]
• the result is achieved by multiplying the exponents:\[a^{m \cdot n}\]

Let's see how this works with an example from our expression: For \[(5x^4y^3)^2\], we apply the Power of a Power Rule to each part inside the parentheses. This gives us:

• \(5^2\)
• \((x^4)^2 = x^{8}\)
• \((y^3)^2 = y^{6}\)

The same logic applies for \[(2x^3y^2)^3\], resulting in:

• \(2^3\)
• \((x^3)^3 = x^9\)
• \((y^2)^3 = y^6\)

This step is crucial as it sets the stage for further simplification.

Once you have simplified using the Power of a Power Rule, you often need to apply the Product of Powers Rule next. This rule is applied when multiplying expressions with the same base. It states that:

• When multiplying like bases, you add the exponents.\[a^m \times a^n = a^{m+n}\]

In our example, we're now faced with multiplying terms from \[5^2x^8y^6 \times 2^3x^9y^6\]to obtain a single expression. First, let's handle the like bases:

• For the \(x\) terms: \(x^8 \times x^9 = x^{17}\)
• For the \(y\) terms: \(y^6 \times y^6 = y^{12}\)

This rule is valuable as it reduces the clutter and makes complex expressions much easier to manage.

Now that we've transformed our expression using the Power of a Power Rule and the Product of Powers Rule, it's time for the final simplification.
Simplification is all about making the equation look cleaner and simpler:- Our ongoing goal was to bring everything together into one neat package.- We started from our expression: \((5x^4y^3)^2(2x^3y^2)^3\)- By using both rules effectively, it has narrowed down to: \[200x^{17}y^{12}\]Simplification often involves some arithmetic too. Up until this point:

• We calculated the coefficients: \(5^2 = 25\) and \(2^3 = 8\)
• Multiplied these coefficients together: \(25 \times 8 = 200\)

And used those results in our final expression. This step pulls together all the hard work. It's where all the magic happens and turns messy math into something much easier to comprehend and use.