Two equations are shown. Equation 1: a^2 = 64. Equation 2: b^3 = 64. Explain why the value of a is different than the value of b using written sentences and match calculations.

Answer 1

To solve the two equations, we can analyze each one separately.

Equation 1: \( a^2 = 64 \)

To find the value of \( a \), we take the square root of both sides:

\[ a = \sqrt{64} \]

Calculating the square root gives us:

\[ a = 8 \quad \text{or} \quad a = -8 \]

This is because squaring either \( 8 \) or \( -8 \) will result in \( 64 \). Thus, the possible values for \( a \) are \( 8 \) and \( -8 \).

Equation 2: \( b^3 = 64 \)

Next, we solve for \( b \) by taking the cube root of both sides:

\[ b = \sqrt[3]{64} \]

Calculating the cube root gives us:

\[ b = 4 \]

In this case, there is only one real cube root. The value of \( 4 \) is the only solution for \( b \) since the cube of a negative number would result in a negative value, and \( 64 \) is positive.

Comparison of Values:

The value of \( a \) can be either \( 8 \) or \( -8 \), while the value of \( b \) is definitively \( 4 \). The main reason these values are different lies in the nature of the operations involved in each equation:

• Squaring (as in Equation 1) can yield both a positive and a negative root, leading to multiple possible values for \( a \).
• Cubing (as in Equation 2) has only one real root, which is positive in this case, resulting in a single value of \( b \).

Thus, the difference in the values of \( a \) and \( b \) arises from the properties of square and cube roots, which allow \( a \) to have two possible values while \( b \) only has one.

Answer 2

can you make this explanation shorter and make it more human like

Answer 3

Sure! Here's a more concise and relatable explanation:

In Equation 1, \( a^2 = 64 \), we find the value of \( a \) by taking the square root of 64, which gives us \( a = 8 \) or \( a = -8 \). This happens because squaring a number can produce both positive and negative results.

In Equation 2, \( b^3 = 64 \), we take the cube root of 64, which gives us \( b = 4 \). Unlike squaring, cubing only has one real solution in this case because it's always positive when dealing with positive numbers.

So, the key difference is that squaring a number can lead to two possible values (like 8 and -8), while cubing gives just one (4). That’s why \( a \) can be different from \( b \).