Understanding the Exponent Simplification Problem

The question asks us to simplify a given algebraic expression involving exponents. Simplifying means rewriting the expression in its simplest possible form. We need to use the rules of exponents to combine the terms effectively.

The expression is: \((x^{\frac{m}{n}})^{m+n} \times (x^{\frac{n}{p}})^{n+p} \times (x^{p} \times x^{m})^{p-m}\)

Let's break down the expression into three parts and simplify each part using exponent rules. 

Key Exponent Rules Used in Simplification

We will be using the following fundamental rules of exponents:

• Quotient Rule: \(\frac{a^b}{a^c} = a^{b-c}\)
• Power of a Power Rule: \((a^b)^c = a^{b \times c}\)
• Product Rule: \(a^b \times a^c = a^{b+c}\)
• Zero Exponent Rule: \(a^0 = 1\) (for \(a \neq 0\))

Step-by-Step Simplification Process

Step 1: Simplify the first term

The first term is \((x^{\frac{m}{n}})^{m+n}\). This looks like a typo in the question formatting (likely meant \((x^m / x^n)^{m+n}\) or similar). Assuming the intent was \((x^m / x^n)^{m+n}\), let's simplify using the Quotient Rule and then the Power of a Power Rule.

Inside the parenthesis: \(\frac{x^m}{x^n} = x^{m-n}\) (using Quotient Rule).

Now apply the outer exponent: \((x^{m-n})^{m+n}\).

Using the Power of a Power Rule: \(x^{(m-n)(m+n)}\).

Recall the difference of squares formula: \((a-b)(a+b) = a^2 - b^2\).

So, \((m-n)(m+n) = m^2 - n^2\).

The first term simplifies to \(x^{m^2 - n^2}\).

Step 2: Simplify the second term

The second term is \((x^{\frac{n}{p}})^{n+p}\). Again, assuming the intent was \((x^n / x^p)^{n+p}\).

Inside the parenthesis: \(\frac{x^n}{x^p} = x^{n-p}\) (using Quotient Rule).

Now apply the outer exponent: \((x^{n-p})^{n+p}\).

Using the Power of a Power Rule: \(x^{(n-p)(n+p)}\).

Using the difference of squares formula: \((n-p)(n+p) = n^2 - p^2\).

The second term simplifies to \(x^{n^2 - p^2}\).

Step 3: Simplify the third term

The third term is \((x^{p} \times x^{m})^{p-m}\).

Inside the parenthesis: \(x^{p} \times x^{m} = x^{p+m}\) (using Product Rule).

Now apply the outer exponent: \((x^{p+m})^{p-m}\).

Using the Power of a Power Rule: \(x^{(p+m)(p-m)}\).

Using the difference of squares formula: \((p+m)(p-m) = p^2 - m^2\).

The third term simplifies to \(x^{p^2 - m^2}\).

Step 4: Multiply the simplified terms

Now we multiply the simplified forms of the three terms:

\(x^{m^2 - n^2} \times x^{n^2 - p^2} \times x^{p^2 - m^2}\)

Using the Product Rule (\(a^b \times a^c = a^{b+c}\)), we add the exponents:

\(x^{(m^2 - n^2) + (n^2 - p^2) + (p^2 - m^2)}\)

Step 5: Simplify the exponent

Let's simplify the sum of the exponents:

\((m^2 - n^2) + (n^2 - p^2) + (p^2 - m^2)\)

\(m^2 - n^2 + n^2 - p^2 + p^2 - m^2\)

Combine like terms:

• \(m^2\) and \(-m^2\) cancel out.
• \(-n^2\) and \(n^2\) cancel out.
• \(-p^2\) and \(p^2\) cancel out.

The sum of the exponents is \(0\).

Step 6: Final Result

The expression simplifies to \(x^0\).

Using the Zero Exponent Rule (\(a^0 = 1\) for \(a \neq 0\)), we conclude that \(x^0 = 1\), assuming \(x \neq 0\).

Thus, the simplified expression is \(1\).

Summary of Simplification Steps

Comparing with Options

The simplified value is \(1\). Let's check the given options:

• mnp
• 0
• 1
• m² - p² + n²

Our result matches option 3.

Revision Table: Exponent Rules Review

Additional Information on Exponents and Simplification

Exponent simplification is a fundamental skill in algebra. It involves rewriting expressions with powers in a more manageable form. These rules are based on the definition of exponents as repeated multiplication.

For example, \(x^3 = x \times x \times x\).

When we multiply \(x^3\) by \(x^2\), we get \((x \times x \times x) \times (x \times x) = x \times x \times x \times x \times x = x^5\). This illustrates the Product Rule: \(x^3 \times x^2 = x^{3+2} = x^5\).

When simplifying complex expressions, always handle operations inside parentheses first, then apply power rules, and finally perform multiplication and division (applying product and quotient rules by adding or subtracting exponents). Remember the order of operations (PEMDAS/BODMAS).

The problem highlighted the importance of recognizing algebraic identities like the difference of squares \((a-b)(a+b) = a^2 - b^2\), which often appears in exponent simplification problems, especially when bases are related by multiplication or division and the outer exponents are sums or differences.

It's crucial to assume that the base \(x\) is not zero, as \(0^0\) is an indeterminate form, though in many contexts involving such algebraic simplifications, the variable is assumed to be a non-zero real number unless stated otherwise.