Question
Solve the equation analytically. (Analytically means with algebraic methods.) Write the answer in exact form using fractions, not decimals. \( \left(2^{18 x+15}\right)\left(2^{x}\right)=32 \) \( x=\square \)
Ask by Brooks Morgan. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -\frac{10}{19} \)
Solution
We start with the given equation:
(2^(18x + 15)) · (2^x) = 32
Using the property of exponents, a^(m) · a^(n) = a^(m+n), combine the left-hand side:
2^(18x + 15 + x) = 2^(19x + 15)
Thus, the equation becomes:
2^(19x + 15) = 32
Recognize that 32 is a power of 2, since 32 = 2^5. Therefore, we set the exponents equal to each other:
19x + 15 = 5
Now, solve for x:
19x = 5 − 15
19x = −10
Divide both sides by 19:
x = −10/19
Thus, the solution in exact fractional form is:
x = −10/19
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Simplify this solution Bonus Knowledge
To solve the equation \( \left(2^{18x+15}\right)\left(2^{x}\right)=32 \), we can first simplify the left side using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). This gives us: \[ 2^{(18x + 15) + x} = 32 \] Combining the exponents results in: \[ 2^{19x + 15} = 32 \] Next, we can express 32 as a power of 2: \( 32 = 2^5 \). This transforms our equation into: \[ 2^{19x + 15} = 2^5 \] Since the bases are the same, we can set the exponents equal to each other: \[ 19x + 15 = 5 \] Now, solving for \( x \): \[ 19x = 5 - 15 \] \[ 19x = -10 \] \[ x = -\frac{10}{19} \] Thus, the solution in exact form is: \( x = -\frac{10}{19} \)
