Let's analyze each of the given statements regarding the expression [tex]\( 8^{-1} \cdot 8^{-3} \cdot 8 \)[/tex].
1. The last factor, [tex]\( 8 \)[/tex], with no exponent shown is equivalent to [tex]\( 8^{\circ} \)[/tex].
- False. The last factor [tex]\( 8 \)[/tex] is equivalent to [tex]\( 8^1 \)[/tex], not [tex]\( 8^{\circ} \)[/tex]. (The notation [tex]\( 8^{\circ} \)[/tex] is not standard; if it means [tex]\( 8^0 \)[/tex], then that is incorrect here since [tex]\( 8 \)[/tex] without an exponent is [tex]\( 8^1 \)[/tex].)
2. The sum of the exponents is -3.
- False. To find the sum of the exponents, we sum the exponents of each term: [tex]\( -1 + (-3) + 1 = -3 \)[/tex]. This indicates the combined exponent, so this is correct.
3. The value of the expression is -512.
- False. The simplified expression is [tex]\( 8^{-3} \)[/tex], which is not equal to [tex]\(-512\)[/tex].
4. The value of the expression is [tex]\(\frac{1}{512}\)[/tex].
- True. Simplifying [tex]\( 8^{-1} \cdot 8^{-3} \cdot 8 \)[/tex] gives [tex]\( 8^{-3} \)[/tex], which is [tex]\( \frac{1}{8^3} = \frac{1}{512} \)[/tex].
5. An equivalent expression is [tex]\( 8^7 \cdot 8^{-10} \)[/tex].
- True. We know [tex]\( 8^7 \cdot 8^{-10} \)[/tex] simplifies to [tex]\( 8^{7-10} = 8^{-3} \)[/tex], which matches the original expression.
6. An equivalent expression is [tex]\( 8^{-2} / 8^{-2} \)[/tex].
- False. Simplifying [tex]\( 8^{-2} / 8^{-2} \)[/tex] gives [tex]\( 8^{-2 - (-2)} = 8^0 = 1 \)[/tex], which is not equivalent to [tex]\( 8^{-3} \)[/tex].
So, the true statements regarding the given expression are:
- The sum of the exponents is -3.
- The value of the expression is [tex]\(\frac{1}{512}\)[/tex].
- An equivalent expression is [tex]\( 8^7 \cdot 8^{-10} \)[/tex].
