If \frac{8^x}{2^y} = 64, which of the following equations correctly expresses y in terms o… — SAT

If $\frac{8^x}{2^y} = 64$ , which of the following equations correctly expresses $y$ in terms of $x$ ?

A
$y = 3x - 64$
$y = 3x - 6$ Cevap
C
$y = 3x + 6$
D
$y = 6 - 3x$

Cevap

y = 3x - 6

To solve the equation, express all terms using the common base of 2. Substituting

8 = 2^3

and

64 = 2^6

into the equation gives

\frac{(2^3)^x}{2^y} = 2^6

. Applying the power rule of exponents transforms the numerator into

2^{3x}

, which leads to the equation

\frac{2^{3x}}{2^y} = 2^6

. Using the quotient rule of exponents, the division simplifies to

2^{3x - y} = 2^6

. Since the bases are now identical, their exponents must be equal:

3x - y = 6

. Isolating the variable yields the correct formulation.

Adım Adım Çözüm

Rewrite all bases in the equation as powers of the common base 2.

Since

8 = 2^3

and

64 = 2^6

, the equation becomes

\frac{(2^3)^x}{2^y} = 2^6

Expressing all exponential terms using a single base allows for the application of exponent rules.

Apply the power of a power property to simplify the numerator on the left side.

The term

(2^3)^x

simplifies to

2^{3x}

, yielding the equation

\frac{2^{3x}}{2^y} = 2^6

The exponent rule

(a^m)^n = a^{mn}

states that a base raised to a power and then another power requires multiplying the exponents.

Apply the quotient property of exponents to combine the left side of the equation.

The left side simplifies to

2^{3x - y}

, resulting in

2^{3x - y} = 2^6

The quotient rule

\frac{a^m}{a^n} = a^{m-n}

states that dividing powers with the same base requires subtracting the exponent of the denominator from the exponent of the numerator.

Set the exponents equal to each other and solve for the desired variable.

3x - y = 6

, which simplifies to

y = 3x - 6

when solved for

y

Since the bases on both sides of the equation are equal, their exponents must also be equal.

Anahtar Kavram

Solving exponential equations by base conversion and rules of exponents

Alternatif Yöntem

Instead of converting both sides to base 2, you can also substitute a test value for

x

. For example, if

x = 2

, then the equation becomes

\frac{8^2}{2^y} = 64

, which simplifies to

\frac{64}{2^y} = 64 \implies 2^y = 1 \implies y = 0

. Substituting

x = 2

into the correct option yields

y = 3(2) - 6 = 0

, confirming the answer.

Tahmini Süre:1m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Konular

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