Video Transcript
Find the value of 𝑥 for which three-fifths to the power of two 𝑥 minus four is equal to two and seven-ninths.
In order to answer this question, we will need to recall some of our laws of exponents. Before doing this, however, we will convert the mixed number two and seven-ninths into an improper or top-heavy fraction. As two is equal to eighteen-ninths, two and seven-ninths is equal to twenty-five ninths. A quick way of converting a mixed number to an improper fraction is to multiply the whole number by the denominator and then add the numerator. Two multiplied by nine plus seven is equal to 25.
We recognize at this point that 25 is five squared and nine is three squared. One of our laws of exponents states that 𝑎 to the power of negative 𝑥 is equal to one over 𝑎 to the power of 𝑥. This means that when dealing with fractions, 𝑎 over 𝑏 to the power of negative 𝑥 is equal to 𝑏 over 𝑎 to the power of 𝑥. This means that three-fifths to the power of negative two is equal to five-thirds squared. Squaring the numerator and denominator gives us 25 over nine or twenty-five ninths.
We can, therefore, rewrite the right-hand side of our equation as three-fifths to the power of negative two. This gives us three-fifths to the power of two 𝑥 minus four is equal to three-fifths to the power of negative two. The exponents must be equal. Two 𝑥 minus four is equal to negative two. Adding four to both sides of this equation gives us two 𝑥 is equal to two. We can then divide both sides of this equation by two, giving us 𝑥 is equal to one.
The value of 𝑥 for which three-fifths to the power of two 𝑥 minus four is equal to two and seven-ninths is one.
