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Simplify five 𝑥 to the negative eight squared times six 𝑥 squared squared.
For this question, we’ll need to utilize a few of the power rules that we know. Here are two of the rules we need to utilize. Taking a power to a power says that to raise a power to a power, you need to multiply the exponents together. And the product rule says that to multiply two exponents with the same base, you keep the base and add the powers.
Let’s start with the power to power rule and distribute our squared. We need to distribute our squared to both the five and the 𝑥 to the negative eight power. We would then have five squared times 𝑥 to the negative eight squared. Five squared is 25 and 𝑥 to the negative eight squared equals 𝑥 to the negative eight times two. So we have 25 times 𝑥 to the negative 16.
Then, we’ll go back up to the second half of our equation and will distribute our two again. From there, we see that we’ll have six squared times 𝑥 squared squared. Six squared equals 36 and 𝑥 squared squared equals 𝑥 to the two times two power; in other words, 𝑥 to the fourth.
To simplify this even further, we can use the product rule and combine our like terms. First, we multiply 25 times 36, which is 900. And then we need to add our exponents because of the product rule. We’ll need to add negative 16 and four to determine what the exponent of our 𝑥 will be. When we add negative 16 and four, we’re left with negative 12. This expression can be simplified to 900 times 𝑥 to the negative 12.
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