Question

What is the greatest integer, x, such that \( \left( \frac{125^x}{25^6} \right)<1 \)?

Hint:

When faced with an exponential inequality, the first step should always be to try to express all terms with a common base. This simplifies the problem from an exponential one to a linear or polynomial one involving just the exponents.

Answer 1

Step 1: Understanding the Concept:
The problem asks for the greatest integer value of \(x\) that satisfies the given exponential inequality. The key to solving this is to simplify the exponential expression by expressing the numbers with a common base and then solving the resulting inequality for the exponents.
Step 2: Key Formula or Approach:
We will use the following rules of exponents:

• Power of a power rule: \((a^m)^n = a^{m \times n}\)
• Quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\)

We will also use the property that any non-zero number raised to the power of 0 is 1 (\(a^0 = 1\)), and for any base \(b > 1\), the inequality \(b^p < b^q\) implies \(p < q\).
Step 3: Detailed Explanation:
We are given the inequality:
\[ \left( \frac{125^x}{25^6} \right) < 1 \]
First, let's express the numbers 125 and 25 as powers of a common base, which is 5:
\[ 125 = 5 \times 5 \times 5 = 5^3 \]
\[ 25 = 5 \times 5 = 5^2 \]
Substitute these into the inequality:
\[ \frac{(5^3)^x}{(5^2)^6} < 1 \]
Apply the power of a power rule \((a^m)^n = a^{mn}\) to both the numerator and the denominator:
\[ \frac{5^{3x}}{5^{12}} < 1 \]
Apply the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\) to the left side:
\[ 5^{3x - 12} < 1 \]
To compare the exponents, we need to express the right side (1) as a power of 5. We know that \(5^0 = 1\).
\[ 5^{3x - 12} < 5^0 \]
Since the base (5) is greater than 1, we can compare the exponents directly without changing the direction of the inequality sign:
\[ 3x - 12 < 0 \]
Now, we solve this linear inequality for \(x\). Add 12 to both sides:
\[ 3x < 12 \]
Divide by 3:
\[ x < 4 \] Step 4: Final Answer:
The inequality \(x < 4\) means that \(x\) can be any number less than 4. The question asks for the greatest integer \(x\) that satisfies this condition. The integers that are less than 4 are 3, 2, 1, 0, -1, and so on. The greatest among these integers is 3.