How to find the solution to a binomial problem - SAT Math

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Question

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

Compare your answer with the correct one above

Question

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

Compare your answer with the correct one above

Question

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

Compare your answer with the correct one above

Question

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

Compare your answer with the correct one above

Question

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

Compare your answer with the correct one above

Question

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

Compare your answer with the correct one above

Question

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

Compare your answer with the correct one above

Question

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

Compare your answer with the correct one above

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How to find the solution to a binomial problem - SAT Math

Define an operation on the set of real numbers as follows: For all real , How else could this operation be defined?

, as the cube of a binomial, can be rewritten using the following pattern: Applying the rules of exponents to simplify this: Therefore, the correct choice is that, alternatively stated, .

Define an operation on the set of real numbers as follows: For all real , How else could this operation be defined?

, as the cube of a binomial, can be rewritten using the following pattern: Applying the rules of exponents to simplify this: Therefore, the correct choice is that, alternatively stated, .

Define an operation on the set of real numbers as follows: For all real , How else could this operation be defined?

, as the cube of a binomial, can be rewritten using the following pattern: Applying the rules of exponents to simplify this: Therefore, the correct choice is that, alternatively stated, .

Define an operation on the set of real numbers as follows: For all real , How else could this operation be defined?

, as the cube of a binomial, can be rewritten using the following pattern: Applying the rules of exponents to simplify this: Therefore, the correct choice is that, alternatively stated, .

Define an operation on the set of real numbers as follows: For all real , How else could this operation be defined?

, as the cube of a binomial, can be rewritten using the following pattern: Applying the rules of exponents to simplify this: Therefore, the correct choice is that, alternatively stated, .

Define an operation on the set of real numbers as follows: For all real , How else could this operation be defined?

, as the cube of a binomial, can be rewritten using the following pattern: Applying the rules of exponents to simplify this: Therefore, the correct choice is that, alternatively stated, .

Define an operation on the set of real numbers as follows: For all real , How else could this operation be defined?

, as the cube of a binomial, can be rewritten using the following pattern: Applying the rules of exponents to simplify this: Therefore, the correct choice is that, alternatively stated, .

Define an operation on the set of real numbers as follows: For all real , How else could this operation be defined?

, as the cube of a binomial, can be rewritten using the following pattern: Applying the rules of exponents to simplify this: Therefore, the correct choice is that, alternatively stated, .

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?