Video Transcript
Find the solution set of two to the power of 𝑥 plus two minus two to the power of 𝑥 minus one is equal to 448 for all real values.
In order to answer this question, we need to recall some of our laws of exponents. Firstly, 𝑎 to the power of 𝑥 plus 𝑦 is equal to 𝑎 to the power of 𝑥 multiplied by 𝑎 to the power of 𝑦. We also recall that 𝑎 to the power of 𝑥 minus 𝑦 is equal to 𝑎 to the power of 𝑥 divided by 𝑎 to the power of 𝑦. We can use these laws of exponents to simplify the two terms on the left-hand side of our equation. Two to the power of 𝑥 plus two is equal to two to the power of 𝑥 multiplied by two squared. As two squared is equal to four, this can be rewritten as four multiplied by two to the power of 𝑥. Two to the power of 𝑥 minus one is equal to two to the power of 𝑥 divided by two to the power of one. This is equal to two to the power of 𝑥 over two, which can be rewritten as a half of two to the power of 𝑥.
Our equation can therefore be rewritten as four multiplied by two to the power of 𝑥 minus a half of two to the power of 𝑥 is equal to 448. Factoring out two to the power of 𝑥, we can subtract one-half from four, which gives us three and a half or seven over two. The equation simplifies to seven over two multiplied by two to the power of 𝑥 is equal to 448. We can then divide both sides of our equation by seven over two or seven-halves. This is the same as multiplying by two-sevenths. The left-hand side simplifies to two to the power of 𝑥, and 448 multiplied by two-sevenths is 128.
We know that two to the seventh power is 128. This means that two to the power of 𝑥 is equal to two to the seventh power. As the bases are now the same, the exponents must also be equal. Therefore, 𝑥 is equal to seven. The solution set of the equation two to the power of 𝑥 plus two minus two to the power of 𝑥 minus one equals 448 is seven. We can check this answer by substituting 𝑥 equals seven back into the equation. Seven plus two is equal to nine. Seven minus one is equal to six. Subtracting two to the sixth power from two to the ninth power does give us 448. This confirms that the correct solution is seven.
