Let x = 2^{100}, y = 3^{75}, and z = 5^{50}. Which of the following inequalities correctly… — ACT

Let $x = 2^{100}$ , $y = 3^{75}$ , and $z = 5^{50}$ . Which of the following inequalities correctly represents the relationship among the values of $x$ , $y$ , and $z$ ?

A
$x < y < z$
$x < z < y$ Answer
C
$z < y < x$
D
$y < z < x$
E
$z < x < y$

Answer

x < z < y

To find the correct relationship, rewrite

x = 2^{100}

y = 3^{75}

, and

z = 5^{50}

with a common exponent. The greatest common divisor of

100

75

, and

50

25

. Using the rule

(a^m)^n = a^{m \cdot n}

, rewrite the terms:

x = (2^4)^{25} = 16^{25}

y = (3^3)^{25} = 27^{25}

, and

z = (5^2)^{25} = 25^{25}

. Comparing the bases shows

16 < 25 < 27

, which means

16^{25} < 25^{25} < 27^{25}

, so

x < z < y

Step-by-Step Solution

Find the greatest common divisor (GCD) of the exponents of the three expressions.

The exponents are

100

75

, and

50

. The greatest common divisor of these numbers is

25

Finding a common exponent allows us to rewrite each expression with the same power so we can compare their bases directly.

Rewrite each expression using the power of a power rule:

(a^m)^n = a^{m \cdot n}

x = 2^{100} = (2^4)^{25}

y = 3^{75} = (3^3)^{25}

, and

z = 5^{50} = (5^2)^{25}

This expresses all three values in the form

b^{25}

, where

b

is the base to be evaluated.

Evaluate the bases inside the parentheses.

2^4 = 16

, so

x = 16^{25}

;

3^3 = 27

, so

y = 27^{25}

;

5^2 = 25

, so

z = 25^{25}

Evaluating the base values simplifies each expression to a single number raised to the power of

25

Compare the evaluated bases and write the resulting inequality.

Since

16 < 25 < 27

, it follows that

16^{25} < 25^{25} < 27^{25}

. This corresponds to

x < z < y

Since the exponent

25

is positive, raising larger positive bases to this exponent results in larger values.

Key Concept

Comparing exponential expressions by rewriting them with a common exponent using exponent rules.

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