Can we use either perfecr square, difference between two squares or distributive law to solve this

Certainly! Let's explore the given problem using the perfect square identity and the difference of squares.

Given:
\[ a + \frac{1}{a} = 8 \]

We need to find:
\[ a^4 + \frac{1}{a^4} \]

First, let's use the perfect square identity to find \( a^2 + \frac{1}{a^2} \).

Start by squaring both sides of the initial equation:
\[ \left( a + \frac{1}{a} \right)^2 = 8^2 \]
\[ a^2 + 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^2} = 64 \]

Since \( a \cdot \frac{1}{a} = 1 \), the equation simplifies to:
\[ a^2 + 2 + \frac{1}{a^2} = 64 \]
\[ a^2 + \frac{1}{a^2} = 62 \]

Next, we will use the difference of squares to find \( a^4 + \frac{1}{a^4} \).

Square \( a^2 + \frac{1}{a^2} \):
\[ \left( a^2 + \frac{1}{a^2} \right)^2 = 62^2 \]
\[ a^4 + 2 \cdot a^2 \cdot \frac{1}{a^2} + \frac{1}{a^4} = 3844 \]

Again, since \( a^2 \cdot \frac{1}{a^2} = 1 \), the equation simplifies to:
\[ a^4 + 2 + \frac{1}{a^4} = 3844 \]
\[ a^4 + \frac{1}{a^4} = 3842 \]

So, using the perfect square identity and the difference of squares, we have:
\[ a^4 + \frac{1}{a^4} = 3842 \]

Thus, the value of \( a^4 + \frac{1}{a^4} \) is \( \boxed{3842} \).

Would you like any further details or have any questions?

Here are 5 additional questions you might ask:
1. How do you solve systems of linear equations using substitution?
2. What are the properties of exponents?
3. How do you find the limit of a function as \( x \) approaches a certain value?
4. Can you explain the binomial theorem?
5. How do you solve a cubic equation?

**Tip:** When dealing with algebraic identities, it's often helpful to recognize and apply relevant formulas such as perfect squares, the difference of squares, and others to simplify your calculations.

Explore how to solve algebraic problems using the perfect square identity and the difference of squares. Learn step-by-step with examples on how to apply these identities effectively.

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