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If a,b, and c are real numbers
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06 Nov 2025, 04:22
06 Nov 2025, 04:22
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If \(a,b\), and \(c\) are real numbers, and $\(\mathrm{a} \neq 0\)$, which of the following must be true?
Indicate all that apply
A. $\(a^*(b+c)=a^* b+a^* c\)$
B. $\(a^*(b-c)=a^* b-a^* c\)$
C. $\((\mathrm{b}+\mathrm{c}) / \mathrm{a}=\mathrm{b} / \mathrm{a}+\mathrm{c} / \mathrm{a}\)$
D. $\((\mathrm{b}-\mathrm{c}) / \mathrm{a}=\mathrm{b} / \mathrm{a}-\mathrm{c} / \mathrm{a}\)$
E. $\(a /(b+c)=a / b+a / c\)$
F. $\(a /(b-c)=a / b-a / c\)$
G. $\((\mathrm{b}+\mathrm{c})^{\mathrm{a}}=\mathrm{b}^{\mathrm{a}}+\mathrm{c}^{\mathrm{a}}\)$
H. $\((b-c)^a=b^a-c^a\)$
Indicate all that apply
A. $\(a^*(b+c)=a^* b+a^* c\)$
B. $\(a^*(b-c)=a^* b-a^* c\)$
C. $\((\mathrm{b}+\mathrm{c}) / \mathrm{a}=\mathrm{b} / \mathrm{a}+\mathrm{c} / \mathrm{a}\)$
D. $\((\mathrm{b}-\mathrm{c}) / \mathrm{a}=\mathrm{b} / \mathrm{a}-\mathrm{c} / \mathrm{a}\)$
E. $\(a /(b+c)=a / b+a / c\)$
F. $\(a /(b-c)=a / b-a / c\)$
G. $\((\mathrm{b}+\mathrm{c})^{\mathrm{a}}=\mathrm{b}^{\mathrm{a}}+\mathrm{c}^{\mathrm{a}}\)$
H. $\((b-c)^a=b^a-c^a\)$
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Official Answer
A,B,C,D
Re: If a,b, and c are real numbers
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10 Dec 2025, 11:07
10 Dec 2025, 11:07
Expert Reply
Distributive properties (A, B)
A. $\(a(b+c)=a b+a c\)$ is the distributive law and is always true for real numbers.
B. $\(a(b-c)=a b-a c\)$ is also always true for real numbers.
So if the symbol is ordinary multiplication, both A and B must be true.
Division over addition/subtraction ( C , D )
C. $\(\frac{b+c}{a}=\frac{b}{a}+\frac{c}{a}\)$ is always true for real $\(a \neq 0\)$, since division by $a$ is just multiplication by $\(\frac{1}{a}\)$, which distributes over addition.
D. $\(\frac{b-c}{a}=\frac{b}{a}-\frac{c}{a}\)$ is likewise always true for real $\(a \neq 0\)$.
So C and D must be true.
Expressions that are not identities ( $\(\mathrm{E}-\mathrm{H}\)$ )
E. $\(\frac{a}{b+c}=\frac{a}{b}+\frac{a}{c}\)$ is not true in general (only in very special cases).
F. $\(\frac{a}{b-c}=\frac{a}{b}-\frac{a}{c}\)$ is also not generally true.
G. $\((b+c)^a=b^a+c^a\)$ fails except for special values (e.g., trivial cases); exponentiation does not distribute over addition.
H. $\((b-c)^a=b^a-c^a\)$ likewise is not generally true.
Therefore, the statements that must be true for all real $\(a \neq 0, b, c\)$ are: $\(\mathrm{A}, \mathrm{B}, \mathrm{C}\)$, and D .
A. $\(a(b+c)=a b+a c\)$ is the distributive law and is always true for real numbers.
B. $\(a(b-c)=a b-a c\)$ is also always true for real numbers.
So if the symbol is ordinary multiplication, both A and B must be true.
Division over addition/subtraction ( C , D )
C. $\(\frac{b+c}{a}=\frac{b}{a}+\frac{c}{a}\)$ is always true for real $\(a \neq 0\)$, since division by $a$ is just multiplication by $\(\frac{1}{a}\)$, which distributes over addition.
D. $\(\frac{b-c}{a}=\frac{b}{a}-\frac{c}{a}\)$ is likewise always true for real $\(a \neq 0\)$.
So C and D must be true.
Expressions that are not identities ( $\(\mathrm{E}-\mathrm{H}\)$ )
E. $\(\frac{a}{b+c}=\frac{a}{b}+\frac{a}{c}\)$ is not true in general (only in very special cases).
F. $\(\frac{a}{b-c}=\frac{a}{b}-\frac{a}{c}\)$ is also not generally true.
G. $\((b+c)^a=b^a+c^a\)$ fails except for special values (e.g., trivial cases); exponentiation does not distribute over addition.
H. $\((b-c)^a=b^a-c^a\)$ likewise is not generally true.
Therefore, the statements that must be true for all real $\(a \neq 0, b, c\)$ are: $\(\mathrm{A}, \mathrm{B}, \mathrm{C}\)$, and D .
gmatclubot
Re: If a,b, and c are real numbers [#permalink]
10 Dec 2025, 11:07
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