Question

Which value is both a squared and a cubed number? (A) 4 (B) 8 (C) 36 (D) 64 (E) 100

Short answer

The value is 64.

Step-by-step solution

- Understanding Squared and Cubed Numbers

A squared number is a number that can be written as the product of an integer with itself, i.e., in the form of \( n^2 \). A cubed number is a number that can be written as an integer multiplied by itself twice, i.e., in the form of \( n^3 \).

- Find the Numbers that are Both Squared and Cubed

A number that is both a squared and cubed number must be in the form of \( n^6 \) because \( (n^2)^3 = n^6 \) and \( (n^3)^2 = n^6 \).

- Check the Given Options

Check each given option to see if it can be written as \( n^6 \):(A) 4 is \( 2^2 \), not \( n^6 \).(B) 8 is \( 2^3 \), not \( n^6 \).(C) 36 is \( 6^2 \), not \( n^6 \).(D) 64 is \( 2^6 \), which is both \( (2^2)^3 \) and \( (2^3)^2 \), thus \( n^6 \).(E) 100 is \( 10^2 \), not \( n^6 \).

- Confirm the Correct Answer

Only 64 can be written both as a squared number \( (2^2)^3 \) and as a cubed number \( (2^3)^2 \). Thus, \( 64 = 2^6 \) and is the correct answer.

Key concepts

Squared Numbers

A squared number, also known as a perfect square, results when an integer is multiplied by itself. For example:

• Example 1: 2 squared is \(2^2 = 4\).
• Example 2: 3 squared is \(3^2 = 9\).

Squaring a number is equivalent to raising it to the power of 2. The formula for a squared number is \(n^2\), where \(n\) is any integer. Square numbers like 4, 9, 16, etc., come from squaring integers.

Cubed Numbers

Cubed numbers are formed by taking an integer and multiplying it by itself twice, written as \(n^3\). This multiplication occurs three times. For example:

• Example 1: 2 cubed is \(2^3 = 8\).
• Example 2: 3 cubed is \(3^3 = 27\).

These cubed numbers growing bigger much faster compared to squared numbers because they involve a higher power.

Exponents

Exponents represent the number of times a base number is multiplied by itself. For example, in \(2^6\), 2 is the base, and 6 is the exponent, meaning \(2 \text {multiplied by itself 6 times}\), which equals 64. Common exponent rules include:

• Multiplication : \((a^m)(a^n) = a^{m+n}\).
• Division : \((a^m)/(a^n) = a^{m-n}\).
• Power of a Power : \((a^m)^n = a^{mn}\).

Recognizing patterns in exponent laws can simplify problems, such as identifying that a number raised to the 6th power (\(n^6\)) can be both a square and a cube.