Solving Exponential Equations: Finding the Value of 'a'

The problem asks us to find the value of 'a' in a given equation involving powers of the number 5. To solve this, we need to simplify both sides of the equation using the rules of exponents.

The given equation is:

$\{(5 \times 5 \times 5 \times 5 \times 5 \times 5)^{5} \times (5 \times 5 \times 5 \times 5 \times 5)^{8}\} \div (5 \times 5) = (625)^{a}$

Step-by-Step Simplification

Let's first rewrite the terms using exponential notation:

• $5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^6$
• $5 \times 5 \times 5 \times 5 \times 5 = 5^5$
• $5 \times 5 = 5^2$
• $625 = 5 \times 5 \times 5 \times 5 = 5^4$

Substitute these exponential forms back into the original equation:

$\{(5^6)^{5} \times (5^5)^{8}\} \div 5^2 = (5^4)^{a}$

Applying Exponent Rules

We will use the following exponent rules:

• Power of a power: $(x^m)^n = x^{m \times n}$
• Product of powers: $x^m \times x^n = x^{m+n}$
• Division of powers: $x^m \div x^n = x^{m-n}$

Let's simplify the left side of the equation first:

Inside the curly braces, we have $(5^6)^5$ and $(5^5)^8$. Apply the power of a power rule:

• $(5^6)^5 = 5^{6 \times 5} = 5^{30}$
• $(5^5)^8 = 5^{5 \times 8} = 5^{40}$

So the equation becomes:

$\{5^{30} \times 5^{40}\} \div 5^2 = (5^4)^{a}$

Now, apply the product of powers rule to the terms inside the curly braces:

• $5^{30} \times 5^{40} = 5^{30 + 40} = 5^{70}$

The equation is now:

$5^{70} \div 5^2 = (5^4)^{a}$

Next, apply the division of powers rule to the left side:

• $5^{70} \div 5^2 = 5^{70 - 2} = 5^{68}$

So the left side simplifies to $5^{68}$.

Now let's simplify the right side of the equation, $(5^4)^{a}$, using the power of a power rule:

• $(5^4)^{a} = 5^{4 \times a} = 5^{4a}$

The equation is now:

$5^{68} = 5^{4a}$

Solving for 'a'

Since the bases on both sides of the equation are equal (both are 5), the exponents must also be equal. Therefore, we can set the exponents equal to each other:

$68 = 4a$

To find the value of 'a', divide both sides by 4:

$a = \frac{68}{4}$

$a = 17$

Thus, the value of 'a' in the equation is 17.

Summary of Simplification Steps

The value of 'a' that satisfies the equation is 17.

Revision Table: Key Exponent Rules

Additional Information: Solving Exponential Equations

An exponential equation is an equation where the variable appears in the exponent. To solve simple exponential equations, we often try to make the bases on both sides of the equation the same. Once the bases are equal, we can equate the exponents and solve the resulting linear equation.

Key steps often involve:

• Rewriting numbers as powers of a common base.
• Using exponent rules to simplify expressions on both sides.
• Equating the exponents once the bases are the same.
• Solving the resulting algebraic equation.

This problem specifically tested the application of the product, quotient, and power rules of exponents.