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In mathematics, the Law of Exponents helps us work with powers or repeated multiplication of numbers. One of these laws is the Power of a Power Law. It states that when you raise a power to another power, like \((a^m)^n\), it equals \(a^{m \times n}\).
This is because multiplying exponents simplifies the process of working with large numbers.Let's delve deeper by applying this law to an example from our exercise. Take the expression \(8^{4/3}\). First, notice that 8 is the same as \(2^3\), since multiplying 2 times itself three times gives us 8.
So, rewriting \(8^{4/3}\) as \((2^3)^{4/3}\) allows us to apply the Power of a Power Law:

• Multiply the exponents: \(3 \times \frac{4}{3} = 4\).
• So \((2^3)^{4/3} = 2^4\).
• This equals 16, since \(2^4 = 16\).

This example shows how the Power of a Power Law can simplify complex expressions by letting you focus only on the multiplication of exponents.

Simplifying expressions requires applying various mathematical rules and laws to make complex mathematical expressions easier to manage. In our example, we applied these concepts to the expression \(x(3x^2)^3\).
A crucial part of simplifying is recognizing the use of exponents and manipulating them using exponent laws.To simplify \((3x^2)^3\):

• Each part of the term \(3x^2\) needs to be raised to the power of 3.
• The coefficient 3 becomes \(3^3\) and the variable \(x^2\) becomes \(x^{2 \times 3}\).
• Calculate \(3^3\) as 27 and \(x^{2 \times 3}\) as \(x^6\).
• This gives us \(27x^6\).

By breaking down the expression into smaller parts and applying the rules, the term \((3x^2)^3\) simplifies to \(27x^6\). The skill of simplifying lets us handle mathematical expressions efficiently and clearly.

The Distributive Property is a fundamental axiom in mathematics which helps in simplifying expressions and performing multiplication effectively. Using this property involves distributing a multiplier to each term within a parenthesis. In our final step, we utilized The Distributive Property to further simplify the expression \(x(3x^2)^3\).
The solution involved several steps:

• We already simplified \((3x^2)^3\) to \(27x^6\).
• Applying the distributive property, we distribute \(x\) across the expression.
• This is done by multiplying \(x\) with the entire term \(27x^6\).
• Using the law \(a^m \cdot a^n = a^{m+n}\), we get \(x^{1+6} = x^7\).
• This simplifies the expression to \(27x^7\).

Through the Distributive Property, simplification reduces the complexity of expressions, enabling easier manipulation and comprehension of mathematical equations.